Μαθηματικά και Στοιχεία Στατιστικής Γ Λυκείου Λύσεις των ασκήσεων

["2.6 f \u2032(x) = 3x2 + 3 = 3(x2 +1) > 0. \u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R.ii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b5\u03af\u03bd\u03b1\u03b9: f \u2032(x) = 6x2 \u2212 6x \u221212 = 6(x2 \u2212 x \u2212 2). \u039f\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03bf\u03c5 \u03c4\u03c1\u03b9\u03c9\u03bd\u03cd\u03bc\u03bf\u03c5 x2 \u2212 x \u2212 2 \u03b5\u03af\u03bd\u03b1\u03b9 2 \u03ba\u03b1\u03b9 \u20131, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u20131 2 +\u221ef \u2032 (x) + 0 \u2013 0+f(x)\u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9:\u2014 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (\u2212\u221e, \u22121], \u03b1\u03c6\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf (\u2212\u221e, \u22121] \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) > 0, \u03c3\u03c4\u03bf (\u2212\u221e, \u22121).\u2014 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [\u22121, 2], \u03b1\u03c6\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [\u22121, 2] \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9f \u2032(x) < 0, \u03c3\u03c4\u03bf (\u22121, 2), \u03ba\u03b1\u03b9\u2014 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [2, +\u221e), \u03b1\u03c6\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [2, +\u221e) \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9f \u2032(x) > 0, \u03c3\u03c4\u03bf (2, +\u221e).iii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9f \u2032(x) = x2 +1\u2212 2x2 = 1\u2212 x2 . (x2 +1)2 (1+ x2 )2\u039f\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03b7\u03c2 f \u2032(x) = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u20131 \u03ba\u03b1\u03b9 1, \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03c4\u03b7\u03c2f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u20131 1 +\u221ef \u2032 (x) \u2013 0 + 0\u2013f ( x)\u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, \u22121], [1, +\u221e) \u03ba\u03b1\u03b9\u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 [\u20131,1]. 100","2.63. i) \u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x < 1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2, \u03c9\u03c2 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae\u2014 \u039f\u03bc\u03bf\u03af\u03c9\u03c2 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 1\u2014 \u0393\u03b9\u03b1 x = 1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5:lim f (x) = lim(4 \u2212 x2 ) = 3, lim f (x) = lim(x + 2) = 3 \u03ba\u03b1\u03b9 f(1) = 3,x \u21921\u2212 x \u21921 x \u21921+ x \u21921\u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf 1.\u0386\u03c1\u03b1 \u03b7 f \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R.\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf R\u2013{1} \u03bc\u03b5 f \u2032( x) = \uf8f1\u22122x, x < 1. \uf8f2 x >1 \uf8f3 1,\u0397 f \u2032(x) = 0 \u03ad\u03c7\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03bc\u03b9\u03b1 \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03bd x = 0. \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 1 +\u221e f \u2032 (x) + 0 \u2013+ f ( x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9:\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, 0] \u03ba\u03b1\u03b9 [1, +\u221e) \u03ba\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0,1].ii) H \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9: \uf8f1x2 \u22121, x \u2208 (\u2212\u221e, \u22121] \uf8f4 f (x) = \uf8f21 \u2212 x2 , x \u2208 (\u22121,1) \uf8f4 x 2 \u22121, x \u2208[1, +\u221e) \uf8f3\u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R, \u03c9\u03c2 \u03b1\u03c0\u03cc\u03bb\u03c5\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c3\u03c5\u03bd\u03b5\u03c7\u03bf\u03cd\u03c2 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7\u03c2.\u25cf \u0393\u03b9\u03b1 x \u2260 \u00b11 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \uf8f1 2x, x \u2208 (\u2212\u221e, \u22121) f \u2032(x) = \uf8f2\uf8f4\u22122x, x \u2208 (\u22121,1). \uf8f4\uf8f3 2x, x \u2208 (1, +\u221e)\u0397 f \u2032 (x) = 0 \u03ad\u03c7\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03bc\u03b9\u03b1 \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03bd x = 0. \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c4\u03b7\u03c2 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. 101","2.6 x \u2212\u221e \u20131 0 1 +\u221e f \u2032 (x) \u2013 + 0\u2013 + f ( x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9:\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, \u22121], [0,1] \u03ba\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 [\u20131,0], [1, +\u221e).4. i) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032( x) = ex \u2212 xex = 1\u2212 x . e2x ex\u0397 f \u2032 (x) = 0 \u03ad\u03c7\u03b5\u03b9 \u03bc\u03b9\u03b1 \u03bc\u03cc\u03bd\u03bf \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03bd x = 1. \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 1 +\u221e f \u2032 (x) + 0 \u2013 f(x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (\u2212\u221e,1] \u03ba\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [1, +\u221e).ii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032(x) = 1 \u22121 = 1\u2212 x . xx \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 f \u2032(x) = 0 \u21d4 x = 1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 1 +\u221e f \u2032 (x) + 0 \u2013 f ( x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (0,1] \u03ba\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [1, +\u221e). 102","2.6iii) \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 f (x) = \uf8f12\u03b7\u00b5x, 0\u2264 x\u2264\u03c0 . \uf8f2\uf8f30, \u03c0 < x \u2264 2\u03c0\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03bd\u03b1 \u03bc\u03b5\u03bb\u03b5\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03c4\u03b7\u03c2 f \u03c3\u03c4\u03bf [0,\u03c0].\u25cf \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [0,\u03c0]\u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0,\u03c0 ) \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032(x) = 2\u03c3\u03c5\u03bdx\u0397 f \u2032 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf (0,\u03c0) \u03b3\u03b9\u03b1 x = \u03c0 . \u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 \u03c3\u03c4\u03bf [0,\u03c0] \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 2\u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u03c0 02\u03c0 f \u2032 (x) + 0 \u2013 f ( x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \uf8ef\uf8ee\uf8f00, \u03c0 \uf8f9 , 2 \uf8fb\uf8fa\u25cf \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \uf8ee\u03c0 , \u03c0 \uf8f9 \u03ba\u03b1\u03b9 \uf8f0\uf8ef 2 \uf8fa\uf8fb\u25cf \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03bc\u03b5 \u03c4\u03b9\u03bc\u03ae \u03bc\u03b7\u03b4\u03ad\u03bd \u03c3\u03c4\u03bf [\u03c0,2\u03c0].5. i) \u25cf \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032(x) = 5x4 + 5 = 5(x4 +1) > 0.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R.\u25cf \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 g \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [0, +\u221e) \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf (0, +\u221e) , \u03bc\u03b5 g\u2032(x) = 2 \u22c5 1 +1 = 1 +1 > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0, +\u221e). 2x x \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 g \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0, +\u221e).ii) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: \u25cf lim f (x) = lim (x5 ) = \u2212\u221e \u03ba\u03b1\u03b9 lim f (x) = lim x5 = +\u221e. x \u2192 \u2212\u221e x \u2192 \u2212\u221e x \u2192 +\u221e x \u2192 +\u221e\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 f , \u03c9\u03c2 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R, \u03b8\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 (\u2212\u221e, +\u221e), \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03c4\u03bf R. 103","2.6\u25cf \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: ( )g(0) = \u2013 3 \u03ba\u03b1\u03b9 lim g(x) = lim 2 x + x \u2212 3 = +\u221e. x \u2192 +\u221e x \u2192 +\u221e\u0386\u03c1\u03b1 \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 g, \u03b3\u03b9\u03b1 \u03c4\u03bf\u03bd \u03af\u03b4\u03b9\u03bf \u03bb\u03cc\u03b3\u03bf \u03cc\u03c0\u03c9\u03c2 \u03c0\u03c1\u03b9\u03bd, \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1[\u22123, +\u221e).iii) \u039f\u0007 \u03b9 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 \u03b3\u03c1\u03ac\u03c6\u03bf\u03bd\u03c4\u03b1\u03b9 f ( x) = 0 \u03ba\u03b1\u03b9 g(x) = 0 \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03af\u03c7\u03c9\u03c2 \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ae \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03bd x = 1. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03bf\u03b9 \u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 f \u03ba\u03b1\u03b9 g \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03bc\u03bf\u03bd\u03cc\u03c4\u03bf\u03bd\u03b5\u03c2, \u03b7 x = 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03ba\u03ae \u03ba\u03bf\u03b9\u03bd\u03ae \u03c1\u03af\u03b6\u03b1 \u03c4\u03bf\u03c5\u03c2.6. i) \u0393\u0007 \u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > \u2013 1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = ex + 1 > 0. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 1+ x\u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (\u22121, +\u221e).ii) \u0397 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 ex = 1\u2212 ln(x +1) \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1: ex \u22121+ ln(x +1) = 0 \u21d4 f (x) = 0.\u03a0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 f(0) = 0. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \u03c0\u03b5\u03b4\u03af\u03bf \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd\u03c4\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f(0) = 0, \u03b7 \u03c4\u03b9\u03bc\u03ae x = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03bc\u03cc\u03bd\u03b7 \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2 f(x) = 0.2.6 \u0392\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0388\u03c3\u03c4\u03c9 x0 \u2208 R. \u03a4\u03cc\u03c4\u03b5, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7\u03c2, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2260 x0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 f (x) \u2212 f (x0 ) \u2264 x \u2212 x0 2 \u21d4 f (x) \u2212 f (x0 ) \u2264 x \u2212 x0 x \u2212 x0 \u21d4\u2212 x \u2212 x0 \u2264 f (x) \u2212 f (x0 ) \u2264 x \u2212 x0 . x \u2212 x0\u0391\u03bb\u03bb\u03ac ( )lim x \u2212 x0 x\u2192 x0 \u2212 x \u2212 x0 = lim = 0. x\u2192 x0\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03ba\u03c1\u03b9\u03c4\u03ae\u03c1\u03b9\u03bf \u03c0\u03b1\u03c1\u03b5\u03bc\u03b2\u03bf\u03bb\u03ae\u03c2 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: lim f (x) \u2212 f (x0 ) = 0 \u03ae f \u2032(x0 ) = 0. x \u2212 x0 x\u2192 x0\u0386\u03c1\u03b1 f \u2032(x0 ) = 0, \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x0 \u2208 R \u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 f \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03ae \u03c3\u03c4\u03bf R. 104","2.62. i) \u0397 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [\u20131,1] \u03c9\u03c2 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = 3x2 \u2212 3 = 3(x2 \u22121) < 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (\u22121,1). \u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [\u20131,1]. ii) \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [\u20131,1], \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf [f(1), f(\u20131)] = [\u03b1 \u2013 2,\u03b1 + 2]. iii) \u0007\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f (x) = x3 \u2212 3x + \u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf [\u20131,1] \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 [\u03b1 \u2013 2,\u03b1 + 2] \u03c0\u03b5\u03c1\u03b9\u03ad\u03c7\u03b5\u03b9 \u03c4\u03bf 0, \u03b1\u03c6\u03bf\u03cd \u2013 2 < \u03b1 < 2. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03c4\u03bf\u03c5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf\u03bd x0 \u2208 (\u22121,1) \u03c4\u03ad\u03c4\u03bf\u03b9\u03bf, \u03ce\u03c3\u03c4\u03b5 f(x0) = 0. \u0391\u03c5\u03c4\u03cc \u03cc\u03bc\u03c9\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03bf\u03bd\u03b1\u03b4\u03b9\u03ba\u03cc, \u03b1\u03c6\u03bf\u03cd \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (\u20131,1).3. \u0397 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b9\u03bd\u03b7\u03c4\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c5(t) = x\u2032(t) = 4t3 \u2212 24t2 + 36t \u221216, \u03b5\u03bd\u03ce \u03b7 \u03b5\u03c0\u03b9\u03c4\u03ac\u03c7\u03c5\u03bd\u03c3\u03ae \u03c4\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1 (t) = x\u2032\u2032(t) = 12t2 \u2212 48t + 36 = 12(t2 \u2212 4t + 3). i) \u0007\u0397 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b9\u03bd\u03b7\u03c4\u03bf\u03cd \u03bc\u03b5 \u03c4\u03b7 \u03b2\u03bf\u03ae\u03b8\u03b5\u03b9\u03b1 \u03c4\u03bf\u03c5 \u03c3\u03c7\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 Horner \u03b3\u03c1\u03ac\u03c6\u03b5\u03c4\u03b1\u03b9 \u03c5(t) = 4(t \u22121)2 (t \u2212 4) \u03ba\u03b1\u03b9 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c4\u03b9\u03c2 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ad\u03c2 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ad\u03c2 t = 1 \u03ba\u03b1\u03b9 t = 4. \u0393\u03b9\u03b1 \u03bd\u03b1 \u03b1\u03c0\u03b1\u03bd\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c3\u03c4\u03b1 \u03b5\u03c1\u03c9\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u03ac\u03c3\u03ba\u03b7\u03c3\u03b7\u03c2 \u03b1\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03bc\u03b5\u03bb\u03b5\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1\u03c2 \u03c5(t) = x\u2032(t) \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 [0,5]. \u039f\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03b7\u03c2 x\u2032(t) = 0 \u03b5\u03af\u03bd\u03b1\u03b9 1 \u03ba\u03b1\u03b9 4, \u03b5\u03bd\u03ce \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 x\u2032(t) \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 t0 1 4 5 x\u2032(t) \u2013 0 \u20130 +ii) \u0386\u0007 \u03c1\u03b1 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 (0,4) \u03c4\u03bf \u03ba\u03b9\u03bd\u03b7\u03c4\u03cc \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03b1\u03c1\u03b9\u03c3\u03c4\u03b5\u03c1\u03ac, \u03b5\u03bd\u03ce \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 (4,5) \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03b4\u03b5\u03be\u03b9\u03ac.iii) \u03a4o \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7\u03c2 \u03b1 (t) = x\u2032\u2032(t) \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 t0 1 3 5 + a(t) + 0 \u2013 0\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 [0,1] \u03ba\u03b1\u03b9 [3,5] \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03ac \u03c4\u03bf\u03c5 \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9, \u03b5\u03bd\u03ce \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1[1,3] \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9. 105","2.64. \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 V \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b3\u03b9\u03b1 t > 0 \u03bc\u03b5 V \u2032(t) = \uf8eb 50 \u2212 25t 2 \uf8f6\u2032 = \u2212 100t < 0 \uf8ec (t + 2)2 \uf8f7 (t + 2)3 \uf8ed \uf8f8\u0386\u03c1\u03b1 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 V \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0, +\u221e), \u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03c4\u03bf\u03c0\u03c1\u03bf\u03ca\u03cc\u03bd \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03c2 \u03c5\u03c0\u03bf\u03c4\u03b9\u03bc\u03ac\u03c4\u03b1\u03b9. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae V(0) = 50 \u03ba\u03b1\u03b9 lim V (t) = \uf8eb 25t 2 \uf8f6 lim \uf8ec 50 \u2212 (t + 2)2 \uf8f7 t \u2192+\u221e \uf8edt \u2192+\u221e \uf8f8 = 50 \u2212 lim t2 25t 2 2 = 50 \u2212 25 = 25, + 4t + t \u2192+\u221e\u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 V \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 (25,50].\u0386\u03c1\u03b1, \u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ca\u03cc\u03bd\u03c4\u03bf\u03c2 \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b3\u03af\u03bd\u03b5\u03b9 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7 \u03b1\u03c0\u03cc \u03c4\u03bf \u03bc\u03b9\u03c3\u03cc \u03c4\u03b7\u03c2 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ae\u03c2\u03c4\u03bf\u03c5 \u03c4\u03b9\u03bc\u03ae\u03c2.5. i) H \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03ad\u03c7\u03b5\u03b9 \u03c0\u03b5\u03b4\u03af\u03bf \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03bf \u0391 = (\u2212\u221e, \u22121) \u222a (\u22121,1) \u222a (1, +\u221e),\u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2, \u03c9\u03c2 \u03c1\u03b7\u03c4\u03ae, \u03ba\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf \u0391 \u03bc\u03b5 f \u2032( x) = (x3 \u2212 9 x)\u2032( x 2 \u22121) \u2212 (x2 \u2212 1)\u2032( x3 \u2212 9x) (x2 \u2212 1) 2 = (3x2 \u2212 9)(x2 \u22121) \u2212 2x(x3 \u2212 9x) (x2 \u22121)2 = x4 + 6x2 + 9 = (x2 + 3)2 > 0. (x2 \u22121)2 (x2 \u22121)2\u0397 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 x \u2212\u221e \u20131 1 +\u221e f \u2032 (x) + + + +\u221e +\u221e +\u221e f(x) \u2212\u221e \u2212\u221e \u2212\u221e\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, \u22121),(\u22121,1) \u03ba\u03b1\u03b9 (1, +\u221e). \u0395\u03af\u03bd\u03b1\u03b9\u25cf lim f (x) = lim x3 \u2212 9x = lim x3 = +\u221e x2 \u22121 x \u2192 +\u221e x2 x \u2192 +\u221e x \u2192 +\u221e 106","2.6\u25cf lim f (x) = lim x3 = \u2212\u221e x \u2192 \u2212\u221e x2 x \u2192 \u2212\u221e\u25cf lim f (x) = lim x3 \u2212 9x = +\u221e x \u2192 \u22121\u2212 x\u2192\u22121\u2212 (x \u22121)(x +1)\u25cf lim f (x) = \u2212\u221e, lim f (x) = +\u221e \u03ba\u03b1\u03b9 lim f (x) = \u2212\u221e. x \u2192 \u22121+ x \u21921\u2212 x \u21921+\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03bf \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 f \u03c3\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03c0 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd\u03c4\u03b7\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf R.ii) \u0007\u039f\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u20131 \u03ba\u03b1\u03b9 1 \u03c0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7\u03c2 x3 \u2212\u03b1 x2 \u2212 9x + \u03b1 = 0. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b8\u03b1 \u03b1\u03bd\u03b1\u03b6\u03b7\u03c4\u03ae\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c1\u03af\u03b6\u03b5\u03c2 \u03b1\u03c5\u03c4\u03ae\u03c2 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, \u22121), (\u22121,1) \u03ba\u03b1\u03b9 (1, +\u221e). \u03a3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ac \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 x3 \u2212\u03b1 x2 \u2212 9x +\u03b1 = 0 \u21d4 x3 \u2212 9x = \u03b1 x2 \u2212\u03b1 \u21d4 x3 \u2212 9x =\u03b1 x2 \u22121 \u21d4 f (x) = \u03b1.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03c3\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03c4\u03c9\u03bd \u03b4\u03b9\u03b1\u03c3\u03c4\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd (\u2212\u221e, \u22121), (\u22121,1) \u03ba\u03b1\u03b9(1, +\u221e) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03ba\u03b1\u03b9 \u03ad\u03c7\u03b5\u03b9 \u03c3\u03cd\u03bd\u03bf\u03bb\u03bf \u03c4\u03b9\u03bc\u03ce\u03bd \u03c4\u03bf R, \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 f(x) =\u03b1, \u03ad\u03c7\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c4\u03c1\u03b5\u03b9\u03c2 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ad\u03c2 \u03c1\u03af\u03b6\u03b5\u03c2, \u03b1\u03c0\u03cc \u03bc\u03b9\u03b1 \u03c3\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1\u03c4\u03bf\u03c5 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03b7\u03c2 f.6. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032(x) = 3\u03b1 x2 + 6x +1. \u0397 f \u2032 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b4\u03b5\u03c5\u03c4\u03b5\u03c1\u03bf\u03b2\u03ac\u03b8\u03bc\u03b9\u03bf \u03c4\u03c1\u03b9\u03ce\u03bd\u03c5\u03bc\u03bf \u03bc\u03b5 \u0394 = 36 \u2013 12\u03b1 = 12(3 \u2013 \u03b1).\u25cf \u0393\u03b9\u03b1 \u03b1 = 3 , \u03b7 f\u2032 \u03ad\u03c7\u03b5\u03b9 \u03b4\u03b9\u03c0\u03bb\u03ae \u03c1\u03af\u03b6\u03b1 \u03c4\u03b7\u03bd \u22121. 3\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03b3\u03b9\u03b1 x = \u2212 1 \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2260 \u2212 1 , \u03b7 f 33\u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R.\u25cf \u0393\u03b9\u03b1 \u03b1 < 3 \u03b7 f \u2032 \u03ad\u03c7\u03b5\u03b9 \u03b4\u03cd\u03bf \u03c1\u03af\u03b6\u03b5\u03c2 \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ad\u03c2 \u03ba\u03b1\u03b9 \u03ac\u03bd\u03b9\u03c3\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf\u03c3\u03c4\u03bf R. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b3\u03b9\u03b1 \u03b1 < 3 \u03b7 f \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R.\u25cf \u0393\u03b9\u03b1 \u03b1 > 3 \u03b7 f \u2032 \u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c3\u03c4\u03bf R \u03ba\u03b1\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b1 > 0 \u03b8\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032 (x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5x \u2208 R. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b3\u03b9\u03b1 \u03b1 > 3 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R.\u0386\u03c1\u03b1 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf R \u03bc\u03cc\u03bd\u03bf \u03cc\u03c4\u03b1\u03bd \u03b1 \u2265 3. 107","2.67. i) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 f \u2032(x) = (\u03b7\u00b5x \u2212 x\u03c3\u03c5\u03bdx)\u2032 = \u03c3\u03c5\u03bdx \u2212 \u03c3\u03c5\u03bdx + x\u03b7\u00b5x = x\u03b7\u00b5x. \u0393\u03b9\u03b1 x \u2208 \uf8eb 0, \u03c0 \uf8f6 \u03b5\u03af\u03bd\u03b1\u03b9 f \u2032 ( x) > 0 \u03ba\u03b1\u03b9 \u03b1\u03c6\u03bf\u03cd \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \uf8ef\uf8ee\uf8f00, \u03c0 \uf8f9 \u03b8\u03b1 \uf8ed\uf8ec 2 \uf8f7\uf8f8 2 \uf8fa\uf8fb \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \u03b1\u03c5\u03c4\u03cc. ii) \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \uf8ee\uf8f0\uf8ef0, \u03c0 \uf8f9 , \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x, \u03bc\u03b5 0< x< \u03c0 \u03b8\u03b1 2 \uf8fa\uf8fb 2 \u03b5\u03af\u03bd\u03b1\u03b9 f(0) < f(x), \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7\u03bcx \u2013 x\u03c3\u03c5\u03bdx > 0. iii) \u0007\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8eb 0, \u03c0 \uf8f6 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \uf8ec\uf8ed 2 \uf8f8\uf8f7 f \u2032( x) = \u03c3\u03c5\u03bdx \u22c5 x \u2212 \u03b7\u00b5x < 0 (\u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 ii), x2 \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 \uf8eb 0, \u03c0 \uf8f7\uf8f8\uf8f6. \uf8ed\uf8ec 28. i) \u0397\u0007 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf \uf8f0\uf8ee\uf8ef0, \u03c0 \uf8f6 , \u03c9\u03c2 \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03bd \u03ba\u03b1\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8eb 0, \u03c0 \uf8f6 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: 2 \uf8f8\uf8f7 \uf8ec\uf8ed 2 \uf8f8\uf8f7 f \u2032( x) = 2\u03c3\u03c5\u03bdx + 1 \u2212 3 = 2\u03c3\u03c5\u03bd3 x \u2212 3\u03c3\u03c5\u03bd2 x +1 \u03c3\u03c5\u03bd2 x \u03c3\u03c5\u03bd2 x = 2\u03c3\u03c5\u03bd3 x \u2212 2\u03c3\u03c5\u03bd2 x \u2212 \u03c3\u03c5\u03bd2 x +1 = 2\u03c3\u03c5\u03bd2 x(\u03c3\u03c5\u03bdx \u22121) \u2212 (\u03c3\u03c5\u03bd2 x \u22121) \u03c3\u03c5\u03bd2 x \u03c3\u03c5\u03bd2 x = (\u03c3\u03c5\u03bdx \u22121)(2\u03c3\u03c5\u03bd2 x \u2212 \u03c3\u03c5\u03bdx \u22121) = (\u03c3\u03c5\u03bdx \u22121)2 (2\u03c3\u03c5\u03bdx + 1) > 0. \u03c3\u03c5\u03bd2 x \u03c3\u03c5\u03bd2 x \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \uf8ee\uf8f0\uf8ef0, \u03c0 \uf8f8\uf8f7\uf8f6. 2 ii) \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf \uf8ef\uf8ee\uf8f00, \u03c0 \uf8f6 , \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 0 \u2264 x < \u03c0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 2 \uf8f8\uf8f7 2 f (0) \u2264 f (x). \u0391\u03bb\u03bb\u03ac f(0) = 0, \u03bf\u03c0\u03cc\u03c4\u03b5 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8ee\uf8f0\uf8ef0, \u03c0 \uf8f6 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: 2 \uf8f8\uf8f7 0 \u2264 2\u03b7\u00b5x + \u03b5\u03d5x \u2212 3x \u21d4 2\u03b7\u00b5x + \u03b5\u03d5x \u2265 3x. 108","","","","","","","","","","","","2.7 \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0, +\u221e), \u03b1\u03c6\u03bf\u03cd \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf 0. \u0386\u03c1\u03b1, \u03b3\u03b9\u03b1 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f(x) > f(0) \u03ae, \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1, (1+ x)\u03bd \u22121\u2212\u03bd x > 0, \u03b1\u03c6\u03bf\u03cd f(0) = (1+0)\u03bd \u2013 1 \u2013 \u03bd\u22190 = 0.\u03b2) \u0391\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9: (1+ x)\u03bd \u22121\u2212\u03bd x \u2212 \u03bd (\u03bd \u22121) x2 > 0. 2 \u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 g(x) = (1+ x)\u03bd \u22121\u2212\u03bd x \u2212\u03bd (\u03bd \u22121) x2 , x \u2265 0. 2 \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 g\u2032(x) =\u03bd (1+ x)\u03bd \u22121 \u2212\u03bd \u2212 \u03bd (\u03bd \u22121) \u22c5 2x 2 = \u03bd (1+ x)\u03bd \u22121 \u2212\u03bd \u2212\u03bd (\u03bd \u22121)x =\u03bd [(1+ x)\u03bd \u22121 \u22121\u2212 (\u03bd \u22121)x] > 0, \u03bb\u03cc\u03b3\u03c9 \u03c4\u03b7\u03c2 \u03b1). \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 g\u2032(x) > 0, \u03b3\u03b9\u03b1 x \u2208 (0, +\u221e) \u03ba\u03b1\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 g \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf 0, \u03b7 g \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0, +\u221e). \u0386\u03c1\u03b1 \u03b3\u03b9\u03b1 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 g(x) > g(0) \u03ae, \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1, (1+ x)\u03bd \u22121\u2212\u03bd x \u2212 \u03bd (\u03bd \u22121) x2 > 0. 24. \u0395\u0007 \u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u2019 \u03cc\u03bb\u03bf \u03c4\u03bf R, \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ae\u03c2 \u03b8\u03b1 \u03b1\u03bd\u03b1\u03b6\u03b7\u03c4\u03b7\u03b8\u03bf\u03cd\u03bd \u03bc\u03cc\u03bd\u03bf \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03c4\u03c9\u03bd \u03c1\u03b9\u03b6\u03ce\u03bd \u03c4\u03b7\u03c2 f \u2032 (x) = 0. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: 6( f (x))2 f \u2032(x) + 6 f \u2032(x) = 6x2 + 6 \u21d4 f \u2032(x)[( f (x))2 +1] = x2 +1 > 0. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 f \u2032 (x) = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b1\u03b4\u03cd\u03bd\u03b1\u03c4\u03b7 \u03c3\u03c4\u03bf R. \u0386\u03c1\u03b1 \u03b7 f \u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1.5. \u0388\u0007 \u03c3\u03c4\u03c9 \u03b1, \u03b2 \u03bf\u03b9 \u03c4\u03b5\u03c4\u03bc\u03b7\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c4\u03c9\u03bd \u03ba\u03bf\u03b9\u03bd\u03ce\u03bd \u03c3\u03b7\u03bc\u03b5\u03af\u03c9\u03bd \u03c4\u03c9\u03bd Cf \u03ba\u03b1\u03b9 Cg. \u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 h(x) = f(x) \u2013 g(x) \u03bc\u03b5 x \u2208[\u03b1 , \u03b2 ], \u03b7 \u03bf\u03c0\u03bf\u03af\u03b1 \u03c0\u03b1\u03c1\u03b9\u03c3\u03c4\u03ac\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c4\u03c9\u03bd Cf \u03ba\u03b1\u03b9 Cg. \u03a4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03be \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c3\u03c9\u03c4\u03b5\u03c1\u03b9\u03ba\u03cc \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 [\u03b1,\u03b2]. \u03a3\u2019 \u03b1\u03c5\u03c4\u03cc \u03b7 h \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ba\u03b1\u03b9 \u03ad\u03c7\u03b5\u03b9 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03bf \u03b8\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 \u03c4\u03bf\u03c5 Fermat \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: 120","2.7 h\u2032(\u03be ) = 0 \u21d4 f \u2032(\u03be ) \u2212 g\u2032(\u03be ) = 0 \u21d4 f \u2032(\u03be ) = g\u2032(\u03be ).\u0386\u03c1\u03b1 \u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391(\u03be, f(\u03be)), \u0392(\u03be, g(\u03be)) \u03bf\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c4\u03c9\u03bd Cf \u03ba\u03b1\u03b9 Cg \u03b1\u03bd\u03c4\u03b9\u03c3\u03c4\u03bf\u03af\u03c7\u03c9\u03c2\u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03ac\u03bb\u03bb\u03b7\u03bb\u03b5\u03c2.6. \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf R \u03bc\u03b5f \u2032(x) = 2(x \u2212\u03b1 )(x \u2212 \u03b2 )2 (x \u2212 \u03b3 )2 + (x \u2212\u03b1 )2 \u22c5 2(x \u2212 \u03b2 )(x \u2212 \u03b3 )2 + (x \u2212\u03b1 )2 (x \u2212 \u03b2 )2 2(x \u2212 \u03b3 ).\u03a0\u03c1\u03bf\u03c6\u03b1\u03bd\u03ce\u03c2 f \u2032(\u03b1 ) = f \u2032(\u03b2 ) = f \u2032(\u03b3 ) = 0. (1)\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03b9\u03ba\u03b1\u03bd\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c4\u03b9\u03c2 \u03c5\u03c0\u03bf\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \u03b8\u03b5\u03c9\u03c1\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 Rolle \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1[\u03b1,\u03b2] \u03ba\u03b1\u03b9 [\u03b2,\u03b3], \u03b1\u03c6\u03bf\u03cd\u25cf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u2019 \u03b1\u03c5\u03c4\u03ac \u03c9\u03c2 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae,\u25cf \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03b1 (\u03b1,\u03b2) \u03ba\u03b1\u03b9 (\u03b2,\u03b3) \u03ba\u03b1\u03b9\u25cf f (\u03b1 ) = f (\u03b2 ) = f (\u03b3 ) = 0.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c5\u03c0\u03ac\u03c1\u03c7\u03bf\u03c5\u03bd \u03be1 \u2208 (\u03b1 , \u03b2 ) \u03ba\u03b1\u03b9 \u03be2 \u2208 (\u03b2 ,\u03b3 ) \u03c4\u03ad\u03c4\u03bf\u03b9\u03b1, \u03ce\u03c3\u03c4\u03b5 f \u2032(\u03be1) = 0 \u03ba\u03b1\u03b9f \u2032(\u03be2 ) = 0. \u0391\u03c0\u03cc (1) \u03ba\u03b1\u03b9 (2) \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 f \u2032 \u03ad\u03c7\u03b5\u03b9 \u03c0\u03ad\u03bd\u03c4\u03b5 \u03c4\u03bf\u03c5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf\u03bd \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03b9\u03c2\u03b1 < \u03be1 < \u03b2 < \u03be2 < \u03b3. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03bb\u03c5\u03c9\u03bd\u03c5\u03bc\u03b9\u03ba\u03ae \u03ad\u03ba\u03c4\u03bf\u03c5 \u03b2\u03b1\u03b8\u03bc\u03bf\u03cd, \u03b7\u03c0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03cc\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03ad\u03bc\u03c0\u03c4\u03bf\u03c5 \u03b2\u03b1\u03b8\u03bc\u03bf\u03cd. \u0386\u03c1\u03b1 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 f \u2032 (x) = 0 \u03b4\u03b5\u03bd \u03ad\u03c7\u03b5\u03b9 \u03ac\u03bb\u03bb\u03b5\u03c2,\u03b5\u03ba\u03c4\u03cc\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b1, \u03be1, \u03b2, \u03be2, \u03b3 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c3\u03c4\u03bf R.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2032 , \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 f \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u03b1 \u03be1 \u03b2 \u03be2 \u03b3 +\u221ef \u2032 (x) \u2013 0 + 0+ 0+f(x) \u03a4.\u0395. 0\u2013 \u03a4.\u0395. 0\u2013 \u03a4.\u0395. \u03a4.\u039c. \u03a4.\u039c.\u0386\u03c1\u03b1 \u03b7 f \u03ad\u03c7\u03b5\u03b9 \u03c4\u03c1\u03af\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ac \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b1 \u03c4\u03b1 f(\u03b1), f(\u03b2) \u03ba\u03b1\u03b9 f(\u03b3) \u03ba\u03b1\u03b9 \u03b4\u03cd\u03bf \u03c4\u03bf\u03c0\u03b9\u03ba\u03ac \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b1\u03c4\u03b1 f(\u03be1) \u03ba\u03b1\u03b9 f(\u03be2).7. i) \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 3x + 4y = 4, \u03bf\u03c0\u03cc\u03c4\u03b5 y = 4 \u2212 3x . 4 121","2.7\u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: \u0395 (x) = \u03951 + \u03952 = x2 3 + y2 4 = x2 3 + \uf8eb 4 \u2212 3x \uf8f62 4 \uf8ec\uf8ed 4 \uf8f7\uf8f8 = x2 3 + 16 \u2212 24x + 9x2 4 16 ( )= 1 \uf8ee 3 x2 \u2212 24x +16\uf8f9\uf8fb . 16 \uf8f0 9+4( )ii) \uf8eb 4 \uf8f6 1 \uf8f0\uf8ee2 x \u2212 24\uf8f9\uf8fb , \u03bf\u03c0\u03cc\u03c4\u03b5\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8ed\uf8ec 0, 3 \uf8f8\uf8f7 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u0395 \u2032(x) = 16 9+4 3 ( ) ( )\u0395 \u2032(x) = 0 \u21d4 x = 12 12 9 \u2212 4 3 4 9\u22124 3 = x1. = = 9 + 4 3 81\u2212 48 11\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032, \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u0395 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd\u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 x1 4 3 \u0395\u2032(x) \u2013 0 + \u0395(x) \u0395(x1)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03c3\u03c7\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03cc\u03c4\u03b1\u03bd \u03b7 \u03c0\u03bb\u03b5\u03c5\u03c1\u03ac \u03c4\u03bf\u03c5( )\u03b9\u03c3\u03bf\u03c0\u03bb\u03b5\u03cd\u03c1\u03bf\u03c5 \u03c4\u03c1\u03b9\u03b3\u03ce\u03bd\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 x = 4 9 \u2212 4 3 \u2245 0, 75 m. 118. i) \u0007\u0388\u03c3\u03c4\u03c9 \u039c(x, f(x)) \u03c4\u03bf \u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03b7\u03c2 Cf . \u0388\u03c7\u03bf\u03c5\u03bc\u03b5(\u039c\u0391)2 = \uf8eb x \u2212 9 \uf8f62 +( f ( x))2 = \uf8eb x \u2212 9 \uf8f62 + x. \uf8ed\uf8ec 2 \uf8f8\uf8f7 \uf8ec\uf8ed 2 \uf8f7\uf8f8\u0397 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7 \u039c\u0391 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b7, \u03cc\u03c4\u03b1\u03bd \u03b3\u03af\u03bd\u03b5\u03b9\u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03cc \u03c4\u03b7\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03cc\u03c4\u03b1\u03bd \u03c0\u03ac\u03c1\u03b5\u03b9\u03c4\u03b7\u03bd \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03b7\u03c2 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 g(x) = \uf8eb x \u2212 9 \uf8f62 + x, x \u2208[0, +\u221e). \uf8ed\uf8ec 2 \uf8f8\uf8f7 122","2.7\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208[0, +\u221e) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 g \u2032( x) = 2 \uf8eb x \u2212 9 \uf8f6 + 1 = 2x \u2212 8, \u03bf\u03c0\u03cc\u03c4\u03b5 \uf8ed\uf8ec 2 \uf8f7\uf8f8 g\u2032(x) = 0 \u21d4 x = 4.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 g\u2032, \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 g \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x 0 4 +\u221e g\u2032(x) \u2013 0 + g(x) 17 4\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 g \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03c4\u03bf x = 4 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf g(4) = 17 . \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 4\u03c0\u03bf\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 (AM)2 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 \u03b7 (AM) \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b7 \u03cc\u03c4\u03b1\u03bd x = 4. \u0386\u03c1\u03b1 \u03c4\u03bf \u03b6\u03b7\u03c4\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u039c(4,2).ii) \u0393\u0007 \u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = 1 , \u03bf\u03c0\u03cc\u03c4\u03b5 \u03bf \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03c3\u03c4\u03ae\u03c2 \u03b4\u03b9\u03b5\u03cd\u03b8\u03c5\u03bd\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 2x\u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7\u03c2 \u03b5 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u039c(4,2) \u03b5\u03af\u03bd\u03b1\u03b9 \u03bb\u03b5 = f \u2032(4) = 1 . \u039f \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03c3\u03c4\u03ae\u03c2\u03b4\u03b9\u03b5\u03cd\u03b8\u03c5\u03bd\u03c3\u03b7\u03c2 \u03c4\u03b7\u03c2 AM \u03b5\u03af\u03bd\u03b1\u03b9: 4 \u03bb\u0391\u039c = 2\u2212 0 = 2 = \u22124. 4\u2212 9 \u22121 22\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03bb\u03b5 \u22c5 \u03bb\u0391\u039c = 1 (\u22124) = \u22121, \u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03b5 \u03b5\u03af\u03bd\u03b1\u03b9 4\u03ba\u03ac\u03b8\u03b5\u03c4\u03b7 \u03c3\u03c4\u03b7\u03bd AM.9. \u0007\u0388\u03c3\u03c4\u03c9 (A\u0392) = 2x \u03ba\u03b1\u03b9 (\u0392\u0393) = y \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b8\u03bf\u03b3\u03c9\u03bd\u03af\u03bf\u03c5 \u0391\u0392\u0393\u0394. \u03a4\u03cc\u03c4\u03b5 \u03b7 \u03c0\u03b5\u03c1\u03af\u03bc\u03b5\u03c4\u03c1\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03c3\u03c4\u03af\u03b2\u03bf\u03c5 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03af\u03c3\u03b7 \u03bc\u03b5 2\u03c0x + 2y \u03ba\u03b1\u03b9 \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b8\u03b1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 2\u03c0 x + 2 y = 400 \u21d4 y = 200 \u2212 \u03c0 x.\u03a4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b8\u03bf\u03b3\u03c9\u03bd\u03af\u03bf\u03c5 A\u0392\u0393\u0394 \u03b5\u03af\u03bd\u03b1\u03b9 \u0395 (x) = 2x \u22c5 y = 2x(200 \u2212 \u03c0 x) = \u22122\u03c0 x2 + 400x. 123","2.7\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8eb 0, 200 \uf8f6 \u03b5\u03af\u03bd\u03b1\u03b9 \u0395\u2032(x) = \u2013 4\u03c0x + 400, \u03bf\u03c0\u03cc\u03c4\u03b5 \uf8ed\uf8ec \u03c0 \uf8f8\uf8f7 \u0395 \u2032(x) = 0 \u21d4 x = 100 . \u03c0\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032, \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u0395 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 100 +\u221e \u03c0 \u0395\u2032(x) + 0 \u2013 max \u0395(x)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 \u0395 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03c4\u03bf x = 100 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf \u0395 \uf8eb 100 \uf8f6 = 20.000 . \u03c0 \uf8ed\uf8ec \u03c0 \uf8f8\uf8f7 \u03c0\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c4\u03bf \u03bf\u03c1\u03b8\u03bf\u03b3\u03ce\u03bd\u03b9\u03bf \u03c4\u03bc\u03ae\u03bc\u03b1 \u03c4\u03bf\u03c5 \u03c3\u03c4\u03af\u03b2\u03bf\u03c5 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf, \u03cc\u03c4\u03b1\u03bd \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2\u03c4\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9: (\u0391\u0392 ) = 2 \u22c5100 = 200 m \u03ba\u03b1\u03b9 (\u0392\u0393 ) = 200 \u2212 \u03c0 \u22c5100 = 100 m. \u03c0\u03c0 \u03c010. \u0007\u0388\u03c3\u03c4\u03c9 x(x > 100) \u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 \u03c4\u03c9\u03bd \u03b1\u03c4\u03cc\u03bc\u03c9\u03bd \u03c0\u03bf\u03c5 \u03b8\u03b1 \u03b4\u03b7\u03bb\u03ce\u03c3\u03bf\u03c5\u03bd \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03bf\u03c7\u03ae. \u03a4\u03cc\u03c4\u03b5, \u03c4\u03bf \u03c0\u03bf\u03c3\u03cc \u03c0\u03bf\u03c5 \u03b8\u03b1 \u03c0\u03bb\u03b7\u03c1\u03ce\u03c3\u03b5\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03ac\u03c4\u03bf\u03bc\u03bf \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03b1\u03bd \u03b1\u03c0\u03cc \u03c4\u03b1 1000 \u03b5\u03c5\u03c1\u03ce \u03b1\u03c6\u03b1\u03b9\u03c1\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c0\u03bf\u03c3\u03cc \u03c4\u03b7\u03c2 \u03ad\u03ba\u03c0\u03c4\u03c9\u03c3\u03b7\u03c2, \u03c4\u03bf \u03bf\u03c0\u03bf\u03af\u03bf \u03b1\u03bd\u03ad\u03c1\u03c7\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5 (x \u2013 100) 5 \u03b5\u03c5\u03c1\u03ce, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03ba\u03ac\u03b8\u03b5 \u03ac\u03c4\u03bf\u03bc\u03bf \u03b8\u03b1 \u03c0\u03bb\u03b7\u03c1\u03ce\u03c3\u03b5\u03b9: 1000 \u2212 (x \u2212100)5 = 1000 \u2212 5x + 500 = 1500 \u2212 5x \u03b5\u03c5\u03c1\u03ce.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c4\u03b1 \u03ad\u03c3\u03bf\u03b4\u03b1 \u03c4\u03b7\u03c2 \u03b5\u03c4\u03b1\u03b9\u03c1\u03b5\u03af\u03b1\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b7 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03bf\u03c7\u03ae \u03c4\u03c9\u03bd x \u03b1\u03c4\u03cc\u03bc\u03c9\u03bd \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9: \u0395 (x) = x(1500 \u2212 5x) = \u22125x2 +1500x.\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 100 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u0395 \u2032(x) = \u221210x +1500, \u03bf\u03c0\u03cc\u03c4\u03b5 \u0395 \u2032(x) = 0 \u21d4 x = 150. \u03a4\u03bf\u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1, \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03bf\u03c0\u03bf\u03af\u03bf \u03c0\u03c1\u03bf\u03c3\u03b4\u03b9\u03bf\u03c1\u03af\u03b6\u03bf\u03c5\u03bc\u03b5\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1\u03c2 \u03c4\u03b7\u03c2 \u0395 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03b1\u03c5\u03c4\u03ae\u03c2. x 100 150 +\u221e \u0395\u2032(x) + 0 \u2013 \u0395(x) 112.500 max 124","2.7\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 \u0395 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03c4\u03bf x0 = 150 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b7 \u03c4\u03b9\u03bc\u03ae \u03c4\u03b7\u03bd \u0395(150) = 112.500.\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b4\u03b7\u03bb\u03ce\u03c3\u03bf\u03c5\u03bd 150 \u03ac\u03c4\u03bf\u03bc\u03b1 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03bf\u03c7\u03ae \u03c3\u03c4\u03b7\u03bd \u03ba\u03c1\u03bf\u03c5\u03b1\u03b6\u03b9\u03ad\u03c1\u03b1 \u03b3\u03b9\u03b1 \u03bd\u03b1\u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c4\u03b1 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1 \u03ad\u03c3\u03bf\u03b4\u03b1.11. \u0007\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 r1\u2032 (t) = 0, 05, \u03bf\u03c0\u03cc\u03c4\u03b5 r1\u2032 (t) = (0, 05t)\u2032 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 r1(t) = 0, 05t + c1. \u038c\u03bc\u03c9\u03c2 r1(0) = 3, \u03bf\u03c0\u03cc\u03c4\u03b5 r1(t) = 0, 05t + 3. \u039f\u03bc\u03bf\u03af\u03c9\u03c2 r2 (t) = 0, 04t + 5. i) \u03a4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03b4\u03b1\u03ba\u03c4\u03c5\u03bb\u03af\u03bf\u03c5 \u03b8\u03b1 \u03bc\u03b7\u03b4\u03b5\u03bd\u03b9\u03c3\u03c4\u03b5\u03af \u03cc\u03c4\u03b1\u03bd r1(t) = r2 (t) \u21d4 3 + 0, 05t = 5 + 0, 04t \u21d4 0, 01t = 2 \u21d4 t = 200. \u0386\u03c1\u03b1, \u03cd\u03c3\u03c4\u03b5\u03c1\u03b1 \u03b1\u03c0\u03cc 200 s \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03b4\u03b1\u03ba\u03c4\u03c5\u03bb\u03af\u03bf\u03c5 \u03b8\u03b1 \u03bc\u03b7\u03b4\u03b5\u03bd\u03b9\u03c3\u03c4\u03b5\u03af. ii) \u03a4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03ba\u03c5\u03ba\u03bb\u03b9\u03ba\u03bf\u03cd \u03b4\u03b1\u03ba\u03c4\u03c5\u03bb\u03af\u03bf\u03c5, \u03c9\u03c2 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c5 t, \u03b5\u03af\u03bd\u03b1\u03b9 \u0395 (t) = \u03c0 r22 (t) \u2212 \u03c0 r12 (t) = \u03c0 (5 + 0, 04t)2 \u2212 \u03c0 (3 + 0, 05t)2.\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u0395 \u2032(t) = 2\u03c0 (5 + 0, 04t) \u22c5 0, 04 \u2212 2\u03c0 (3 + 0, 05t) \u22c5 0, 05 = 2\u03c0 (0, 20 + 0, 0016t \u2212 0,15 \u2212 0, 0025t) = 2\u03c0 (0, 05 \u2212 0, 0009t).\u0395\u03af\u03bd\u03b1\u03b9 \u0395 \u2032(t) = 0 \u21d4 t \u2248 55, 6 s.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032, \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u0395 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. t0 55,6 \u0395\u2032(t) +0 \u2013 \u0395(t) max \u0386\u03c1\u03b1, \u03c4\u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c3\u03c4\u03b9\u03b3\u03bc\u03ae t \u2248 55, 6 s \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03ba\u03c5\u03ba\u03bb\u03b9\u03ba\u03bf\u03cd \u03b4\u03b1\u03ba\u03c4\u03c5\u03bb\u03af\u03bf\u03c5 \u03b8\u03b1 \u03bc\u03b5\u03b3\u03b9\u03c3\u03c4\u03bf\u03c0\u03bf\u03b9\u03b7\u03b8\u03b5\u03af.12. i) \u0007\u0397 \u03ba\u03ac\u03b8\u03b5\u03c4\u03b7 \u03b4\u03b9\u03b1\u03c4\u03bf\u03bc\u03ae \u0391\u0392\u0393\u0394 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c7\u03ae\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03c1\u03b1\u03c0\u03b5\u03b6\u03af\u03bf\u03c5. \u0391\u03c0\u03cc \u03c4\u03bf \u03c4\u03c1\u03af\u03b3\u03c9\u03bd\u03bf \u0397\u0392\u0393 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u0397\u0392 = 2\u03b7\u03bc\u03b8 \u03ba\u03b1\u03b9 \u0397\u0393 = 2\u03c3\u03c5\u03bd\u03b8. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03bf \u03c4\u03c1\u03b1\u03c0\u03ad\u03b6\u03b9\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03b9\u03c3\u03bf\u03c3\u03ba\u03b5\u03bb\u03ad\u03c2, \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 125","2.7 \u0394\u0398 = \u0393\u0397 = 2\u03c3\u03c5\u03bd\u03b8 \u03ba\u03b1\u03b9 \u0394\u0393 = 2 + 2 \u03c3\u03c5\u03bd\u03b8 + 2\u03c3\u03c5\u03bd\u03b8 = 2 + 4 \u03c3\u03c5\u03bd\u03b8.\u03a4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03c4\u03bf\u03c5 \u03c4\u03c1\u03b1\u03c0\u03b5\u03b6\u03af\u03bf\u03c5 \u0391\u0392\u0393\u0394 \u03b5\u03af\u03bd\u03b1\u03b9 E = \u0391\u0392 + \u0393\u2206 \u22c5 \u0397\u0392 = 2 + 2 + 4\u03c3\u03c5\u03bd\u03b8 \u22c5 2\u03b7\u00b5\u03b8 22 = (4 + 4\u03c3\u03c5\u03bd\u03b8 )\u03b7\u00b5\u03b8 = 4\u03b7\u00b5\u03b8 (1+ \u03c3\u03c5\u03bd\u03b8 ).ii) \u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u0395 (\u03b8 ) = 4\u03b7\u00b5\u03b8 (1+ \u03c3\u03c5\u03bd\u03b8 ), \u03b8 \u2208 \uf8eb 0, \u03c0 \uf8f9 . \uf8ed\uf8ec 2 \uf8fb\uf8fa\u0395\u03af\u03bd\u03b1\u03b9 \u0395 \u2032(\u03b8 ) = 4\u03c3\u03c5\u03bd\u03b8 (1+ \u03c3\u03c5\u03bd\u03b8 ) + 4\u03b7\u00b5\u03b8 (\u2212\u03b7\u00b5\u03b8 ) = 4\u03c3\u03c5\u03bd2\u03b8 \u2212 4\u03b7\u00b52\u03b8 + 4\u03c3\u03c5\u03bd\u03b8 = 4\u03c3\u03c5\u03bd2\u03b8 \u2212 4(1\u2212 \u03c3\u03c5\u03bd2\u03b8 ) + 4\u03c3\u03c5\u03bd\u03b8 = 8\u03c3\u03c5\u03bd2\u03b8 + 4\u03c3\u03c5\u03bd\u03b8 \u2212 4 = 4(2\u03c3\u03c5\u03bd2\u03b8 + \u03c3\u03c5\u03bd\u03b8 \u22121).\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u0395 \u2032(\u03b8 ) = 0 \u21d4 2\u03c3\u03c5\u03bd2\u03b8 + \u03c3\u03c5\u03bd\u03b8 \u22121 = 0 \u21d4 \u03c3\u03c5\u03bd\u03b8 = 1 \u03ae \u03c3\u03c5\u03bd\u03b8 = \u22121 2 \u21d4\u03b8 = \u03c0 , \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b8 \u2208 \uf8eb 0, \u03c0 \uf8f9 . 3 \uf8ed\uf8ec 2 \uf8fa\uf8fb\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u0395\u2032 \u03ba\u03b1\u03b8\u03ce\u03c2 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u0395 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. \u03b8 0 \u03c0\/3 \u03c0 \u0395\u2032(\u03b8) +0 \u2013 \u0395(\u03b8) max\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03cc\u03c4\u03b1\u03bd \u03b8 = \u03c0 \u03c4\u03bf \u03b5\u03bc\u03b2\u03b1\u03b4\u03cc\u03bd \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf, \u03c0\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03c4\u03cc\u03c4\u03b5 \u03c4\u03bf 3\u03ba\u03b1\u03bd\u03ac\u03bb\u03b9 \u03b8\u03b1 \u03bc\u03b5\u03c4\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03c4\u03b7 \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03b7 \u03c0\u03bf\u03c3\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bd\u03b5\u03c1\u03bf\u03cd. 126","2.713. i) \u0007\u0388\u03c3\u03c4\u03c9 t1 \u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bf \u03ba\u03bf\u03bb\u03c5\u03bc\u03b2\u03b7\u03c4\u03ae\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03ba\u03bf\u03bb\u03c5\u03bc\u03c0\u03ae\u03c3\u03b5\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u039a \u03c3\u03c4\u03bf \u039c \u03ba\u03b1\u03b9 t2 \u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03c0\u03bf\u03c5 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c0\u03b5\u03c1\u03c0\u03b1\u03c4\u03ae\u03c3\u03b5\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf \u039c \u03c3\u03c4\u03bf \u03a3. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 t1 = (KM ) = x2 +1002 \u03ba\u03b1\u03b9 t2 = (\u039c\u03a3 ) = 300 \u2212 x. \u03c51 3 \u03c52 5\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03bf \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc\u03c2 \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03bd\u03cd\u03c3\u03b5\u03b9 \u03c4\u03b7 \u03b4\u03b9\u03b1\u03b4\u03c1\u03bf\u03bc\u03ae \u039a\u039c\u03a3 \u03b5\u03af\u03bd\u03b1\u03b9 \u03a4 (x) = x2 +1002 + 300 \u2212 x . 35ii) \u0398\u03b5\u03c9\u03c1\u03bf\u03cd\u03bc\u03b5 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03a4 (x) = x2 +1002 + 300 \u2212 x , x \u2208 (0,300). 35\u0395\u03af\u03bd\u03b1\u03b9 \u03a4 \u2032(x) = 1 \u22c5 x \u2212 1 . 3 x2 +1002 5\u039f\u03b9 \u03c1\u03af\u03b6\u03b5\u03c2 \u03c4\u03b7\u03c2 \u03a4\u2032(x) = 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf 75.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03a4\u2032 \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u03a4 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 75 300 \u03a4 \u2032 (x) \u2013 0+ \u03a4(x) \u03a4(75) min\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03a4 \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03b3\u03b9\u03b1 x = 75 ft.\u0386\u03c1\u03b1, \u03cc\u03c4\u03b1\u03bd x = 75ft, \u03c4\u03cc\u03c4\u03b5 \u03bf \u03ba\u03bf\u03bb\u03c5\u03bc\u03b2\u03b7\u03c4\u03ae\u03c2 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c4\u03bf \u03bb\u03b9\u03b3\u03cc\u03c4\u03b5\u03c1\u03bf \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc \u03c7\u03c1\u03cc\u03bd\u03bf\u03b3\u03b9\u03b1 \u03bd\u03b1 \u03c6\u03b8\u03ac\u03c3\u03b5\u03b9 \u03c3\u03c4\u03bf \u03c3\u03c0\u03af\u03c4\u03b9 \u03c4\u03bf\u03c5.14. \u0388\u0007 \u03c3\u03c4\u03c9 \u03c11 \u03b7 \u03c0\u03c5\u03ba\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b1\u03c0\u03bd\u03bf\u03cd \u03c0\u03bf\u03c5 \u03b5\u03ba\u03c0\u03ad\u03bc\u03c0\u03b5\u03b9 \u03c4\u03bf \u03b5\u03c1\u03b3\u03bf\u03c3\u03c4\u03ac\u03c3\u03b9\u03bf \u03951 \u03ba\u03b1\u03b9 \u03c12 \u03b7 \u03c0\u03c5\u03ba\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b1\u03c0\u03bd\u03bf\u03cd \u03c0\u03bf\u03c5 \u03b5\u03ba\u03c0\u03ad\u03bc\u03c0\u03b5\u03b9 \u03c4\u03bf \u03b5\u03c1\u03b3\u03bf\u03c3\u03c4\u03ac\u03c3\u03b9\u03bf \u03952. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u03c11 ( x) = k P \u03ba\u03b1\u03b9 \u03c12 (x) = k 8P , k \u2208R. x2 (12 \u2212 x)2 127","2.7\u0397 \u03c0\u03c5\u03ba\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b1\u03c0\u03bd\u03bf\u03cd \u03c3\u03c4\u03b7 \u03b8\u03ad\u03c3\u03b7 \u03a3 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c1(x) = \u03c11(x) + \u03c12 (x) = k P + k 8P x2 (12 \u2212 x)2 = kP \uf8eb 1 + 8 x)2 \uf8f7\uf8f6. \uf8ec x2 (12 \u2212 \uf8f8 \uf8ed\u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c1(x) = kP \uf8eb 1 + 8 x)2 \uf8f6 , x \u2208 (0,12) \uf8ec x2 (12 \u2212 \uf8f7 \uf8ed \uf8f8\u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03bc\u03b5 \u03c1 \u2032( x) = kP \uf8eb \u2212 2x + 16(12 \u2212 x) \uf8f6 \uf8ec x4 (12 \u2212 x)4 \uf8f7 \uf8ed \uf8f8 = kP \uf8eb \u2212 2 + 16 \uf8f7\uf8f6. \uf8ec x3 (12 \u2212 x)3 \uf8f8 \uf8ed\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \u22122 16 x3 (12 \u2212 x)3 \u03c1 \u2032( x) =0 \u21d4 + =0 \u21d4 16x3 \u2212 2(12 \u2212 x)3 = 0 \u21d4 (2x)3 \u2212 (12 \u2212 x)3 = 0 \u21d4 2x \u2212 (12 \u2212 x) = 0 \u21d4 3x = 12 \u21d4 x = 4.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03c1\u2032, \u03b7 \u03bc\u03bf\u03bd\u03bf\u03c4\u03bf\u03bd\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03c4\u03b7\u03c2 \u03c1 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9\u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 4 12 \u03c1\u2032(x) \u2013 0 + \u03c1(x) min\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 \u03c0\u03c5\u03ba\u03bd\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c1 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03b7, \u03cc\u03c4\u03b1\u03bd x = 4.\u0386\u03c1\u03b1, \u03bf \u03b5\u03c1\u03b3\u03bf\u03bb\u03ac\u03b2\u03bf\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03ad\u03c7\u03b5\u03b9 \u03c4\u03b7 \u03bb\u03b9\u03b3\u03cc\u03c4\u03b5\u03c1\u03b7 \u03c1\u03cd\u03c0\u03b1\u03bd\u03c3\u03b7 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03c7\u03c4\u03af\u03c3\u03b5\u03b9 \u03c4\u03bf \u03c3\u03c0\u03af\u03c4\u03b9 \u03c4\u03bf\u03c5 \u03c3\u03b5\u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7 4 km \u03b1\u03c0\u03cc \u03c4\u03bf \u03b5\u03c1\u03b3\u03bf\u03c3\u03c4\u03ac\u03c3\u03b9\u03bf \u03951. 128","2.82.8 \u0391\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. i) \u0007\u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = 15x4 \u2212 20x3 \u03ba\u03b1\u03b9 f \u2032\u2032(x) = 60x3 \u2212 60x2 = 60x2 (x \u22121),\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = 0 (\u03b4\u03b9\u03c0\u03bb\u03ae) \u03ae x = 1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 1 +\u221e +f \u2033(x) \u20130\u2013 0 f ( x) \u03a3.\u039a.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03bf\u03af\u03bb\u03b7 \u03c3\u03c4\u03bf (\u2212\u221e, 0] \u03ba\u03b1\u03b9 \u03c3\u03c4\u03bf [0,1] \u03ba\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03c3\u03c4\u03bf [1, +\u221e).\u25cf \u03a4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b8\u03ad\u03c3\u03b7 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(1,0) \u03b5\u03af\u03bd\u03b1\u03b9\u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cf .ii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R * \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: g \u2032( x) = 6x4 \u2212 3x2 (3x2 \u2212 2) = 6 \u2212 3x2 x6 x4\u03ba\u03b1\u03b9 g \u2032\u2032( x) = \u22126 x5 \u2212 4x3 (6 \u2212 3x2 ) = 6(x2 \u2212 4) ,\u03bf\u03c0\u03cc\u03c4\u03b5 x8 x5 g\u2032\u2032(x) = 0 \u21d4 x = \u22122 \u03ae x = 2.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 g\u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u20132 0 2 +\u221e 0 +\u2013 0 + g\u2033(x) \u2013 g(x) \u03a3.\u039a. \u03a3.\u039a.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 g \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 [\u20132,0) \u03ba\u03b1\u03b9 [2, +\u221e),\u03b5\u03bd\u03ce \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, \u22122] \u03ba\u03b1\u03b9 (0,2]. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 g\u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9\u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u20132, 2 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391 \uf8eb \u22122, \u22125 \uf8f6 \u03ba\u03b1\u03b9 \uf8ec\uf8ed 4 \uf8f7\uf8f8 129","2.8\u0392 \uf8eb 2, 5 \uf8f6 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cg. \uf8ed\uf8ec 4 \uf8f8\uf8f72. i) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = e1\u2212x \u2212 xe1\u2212x \u03ba\u03b1\u03b9 f \u2032\u2032(x) = e1\u2212x (x \u2212 2),\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = 2.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 2 +\u221e f \u2033(x) \u2013 0+ f ( x) 2 e \u03a3.\u039a.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf (\u2212\u221e, 2] \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9\u03c3\u03c4\u03bf [2, +\u221e).\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 2 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf\u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391 \uf8eb 2, e \uf8f6 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cf . \uf8ec\uf8ed 2 \uf8f7\uf8f8ii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (0, +\u221e) \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9:\u03ba\u03b1\u03b9 g\u2032(x) = 2x(2 ln x \u2212 5) + 2x = 4x(ln x \u2212 2)\u03bf\u03c0\u03cc\u03c4\u03b5 g\u2032\u2032(x) = 4(ln x \u2212 2) + 4 = 4(ln x \u22121), g\u2032\u2032(x) = 0 \u21d4 ln x = 1 \u21d4 x = e.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 g\u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x0 e +\u221e g\u2033(x) \u20130 + g(x) \u22123e2 \u03a3.\u039a. 130","2.8\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 g \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 (0,e] \u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9\u03c3\u03c4\u03bf [e, +\u221e) . \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 g\u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf e \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9\u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf A(e,\u20133e2) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cg.iii) \u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x < 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 h\u2032(x) = \u2013 6x. \u2014 \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x > 0 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 h\u2032(x) = \u2013 3x2 + 6x \u2014 \u03a3\u03c4\u03bf x = 0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: lim h(x) \u2212 h(0) = lim \u22123x2 +1\u22121 = 0 \u03ba\u03b1\u03b9 x \u2192 0\u2212 x x \u2192 0\u2212 x lim h(x) \u2212 h(0) = lim \u2212x3 + 3x2 +1\u22121 = lim (x(3 \u2212 x)) = 0. x \u2192 0+ x x \u2192 0+ x x \u2192 0+\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 h \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf x = 0 \u03bc\u03b5 h\u2032(0) = 0. \u0386\u03c1\u03b1 h\u2032( x) = \uf8f1 \u2212 6x, x, x<0 \uf8f3\uf8f2\u22123x2 + 6 . x\u22650\u0393\u03b9\u03b1 x\u2260 0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 h\u2032\u2032( x) = \uf8f1 \u2212 6, x < 0 , \uf8f2\uf8f3\u22126x + 6, x > 0\u03bf\u03c0\u03cc\u03c4\u03b5 h\u2032\u2032(x) = 0 \u21d4 x = 1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 h\u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 1 +\u221e h\u2033(x) \u2013 +0\u2013 h(x) 13 \u03a3.\u039a. \u03a3.\u039a.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 h \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 (\u2212\u221e, 0] \u03ba\u03b1\u03b9 [1, +\u221e)\u03ba\u03b1\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf [0,1].\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03bf 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c3\u03c9\u03c4\u03b5\u03c1\u03b9\u03ba\u03cc \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03bf\u03c5 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03b7\u03c2 h \u03ba\u03b1\u03b9 h\u2032(0) = 0, \u03b7Ch \u03ad\u03c7\u03b5\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf (0,1) \u03ba\u03b1\u03b9 \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 h\u2033 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03c4\u03bf\u03c5 0 \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9\u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(0,1) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Ch.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 h\u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf 1 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03bf\u03cd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf\u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0392(1,3) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Ch. 131","2.83. i) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = \u22122xe\u2212x2 \u03ba\u03b1\u03b9 f \u2032\u2032(x) = \u22122e\u2212x2 \u2212 2x(e\u2212x2 )\u2032 = 2e\u2212x2 (2x2 \u22121), \u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = 2 \u03ae x = \u2212 2 . 22\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u22122 2 +\u221e 2 2 f \u2033(x) +0 \u2013 0+ f(x) \u03a3.\u039a. \u03a3.\u039a.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03b5 \u03ba\u03b1\u03b8\u03ad\u03bd\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b1 \u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1\uf8eb 2\uf8f9 \u03ba\u03b1\u03b9 \uf8ee 2 , +\u221e \uf8f6 , \u03b5\u03bd\u03ce \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1\uf8ed\uf8ec\uf8ec \u2212\u221e, \u2212 \uf8fa \uf8ef 2 \uf8f8\uf8f7\uf8f7 2 \uf8fb \uf8f0\uf8ee 2, 2 \uf8f9 .\uf8ef\u2212 2 2 \uf8fa\uf8f0 \uf8fb\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f\u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u2212 2 , 2 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03ce\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 22 \uf8eb 2 \u22121 \uf8f6 \u0392 \uf8eb 2 , \u22121 \uf8f6 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cf .\u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391 \uf8ec\uf8ed\uf8ec \u2212 2 \uf8f8\uf8f7\uf8f7 , \uf8ec\uf8ec\uf8ed 2 \uf8f8\uf8f7\uf8f7 ,e 2 e2ii) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 \uf8eb \u2212 \u03c0 , \u03c0 \uf8f6 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 g \u2032( x) = 1 \u03ba\u03b1\u03b9 \uf8ec\uf8ed 2 2 \uf8f7\uf8f8 \u03c3\u03c5\u03bd2 x g \u2032\u2032( x) = \u2212 2\u03c3\u03c5\u03bdx(\u2212\u03b7\u00b5x) = 2\u03b7\u00b5x , \u03bf\u03c0\u03cc\u03c4\u03b5 \u03c3\u03c5\u03bd4 x \u03c3\u03c5\u03bd3 x g\u2032\u2032(x) = 0 \u21d4 \u03b7\u00b5x = 0 \u21d4 x = 0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 g\u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2013\u03c0\/2 0 \u03c0\/2 g\u2033(x) \u2013 0 g(x) + \u03a3.\u039a. 132","2.8 \u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 g \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf \uf8eb \u2212 \u03c0 , 0\uf8fa\uf8fb\uf8f9 , \u03b5\u03bd\u03c9 \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf \uf8ed\uf8ec 2 \uf8ef\uf8ee\uf8f00, \u03c0 \uf8f8\uf8f7\uf8f6. 2 \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 g\u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 0 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03bf\u03cd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u039f(0,0) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cg.iii) \u0395\u03af\u03bd\u03b1\u03b9 h(x) = \uf8f1\uf8f4 x2 , x \u2265 0. \uf8f4\uf8f3\uf8f2\u2212x2 , x<0 \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 h \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R \u03c9\u03c2 \u03b3\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03bf \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03bd. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: h\u2032( x) = \uf8f1 2x, x>0 \u03ba\u03b1\u03b9 h\u2032\u2032( x) = \uf8f1 2, x > 00. \uf8f2\uf8f3\u22122x, x<0 \uf8f3\uf8f2\u22122, x < \u0393\u03b9\u03b1 x0 = 0 \u03b5\u03af\u03bd\u03b1\u03b9 lim h(x) \u2212 h(0) = lim x2 \u2212 0 = 0, x \u2192 0+ x xx \u2192 0+ lim h(x) \u2212 h(0) = lim \u2212x2 = 0. x \u2192 0\u2212 x xx \u2192 0\u2212 \u0386\u03c1\u03b1 h\u2032(0) = 0. \u0391\u03c0\u03cc \u03c4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 h\u2033 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 h \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03c3\u03c4\u03bf [0, +\u221e), \u03ba\u03bf\u03af\u03bb\u03b7 \u03c3\u03c4\u03bf (\u2212\u221e, 0] \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u039f(0,0) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2.iv) \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c6 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf R \u03c9\u03c2 \u03c3\u03cd\u03bd\u03b8\u03b5\u03c3\u03b7 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03bd. \u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \uf8f1 1, x>0 \uf8f4\uf8f4 2 x x<0 \u03d5 (x) = \uf8f1\uf8f4 x, x \u2265 0, \u03d5 \u2032( x) = \uf8f2 1 \uf8f2 \uf8f4\uf8f4\uf8f3\u2212 \uf8f4\uf8f3 \u2212x, x < 0 2\u2212 , x \u03ba\u03b1\u03b9 \u03d5 \u2032\u2032( x) = \uf8f1 4 \u22121 , x>0 \uf8f4\uf8f4 4 xx . \uf8f2 , \uf8f4 1 x<0 \uf8f3\uf8f4 x \u2212x \u0397 \u03c0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c6 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 0 \u03b8\u03b1 \u03b1\u03bd\u03b1\u03b6\u03b7\u03c4\u03b7\u03b8\u03b5\u03af \u03bc\u03b5 \u03c4\u03b7 \u03b2\u03bf\u03ae\u03b8\u03b5\u03b9\u03b1 \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd. 133","2.8\u2014 \u0393\u03b9\u03b1 x > 0, \u03b5\u03af\u03bd\u03b1\u03b9 lim \u03d5(x) \u2212\u03d5(0) = lim x = lim 1 = +\u221e. x\u21920+ x \u2212 0 xx \u2192 0+ xx \u2192 0+\u2014 \u0393\u03b9\u03b1 x < 0, \u03b5\u03af\u03bd\u03b1\u03b9 lim \u03d5(x) \u2212\u03d5(0) = lim \u2212x = lim \u22121 = \u2212\u221e. x\u21920\u2212 x \u2212 0 xx \u2192 0\u2212 x\u21920\u2212 \u2212 x\u0386\u03c1\u03b1, \u03b7 \u03c6 \u03b4\u03b5\u03bd \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf 0. \u038c\u03bc\u03c9\u03c2 \u03b7 C\u03c6 \u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf\u039f(0,\u03c6(0)), \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 x = 0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03c6\u2033, \u03ba\u03b1\u03b8\u03ce\u03c2 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03ba\u03c5\u03c1\u03c4\u03ac \u03c4\u03b7\u03c2 \u03c6 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 +\u221e \u03c6\u2033(x) \u2013 \u2013 \u03c6(x) 0\u0386\u03c1\u03b1 \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u039f(0,0) \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 C\u03c6 , \u03b1\u03c6\u03bf\u03cd \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03c4\u03bf\u03c50 \u03b7 \u03c6\u2033 \u03b4\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf.v) \u0397 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03c8 \u03b3\u03b9\u03b1 x < 0 \u03ba\u03b1\u03b9 \u03b3\u03b9\u03b1 x > 0 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c9\u03c2 \u03c3\u03cd\u03bd\u03b8\u03b5\u03c3\u03b7 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ce\u03bd.\u0399\u03c3\u03c7\u03cd\u03b5\u03b9 ( )lim\u03c8 (x) = lim \u2212 \u2212x = 0, x \u2192 0\u2212 x \u2192 0\u2212 lim \u03c8 (x) = lim x = 0 \u03ba\u03b1\u03b9 \u03c8(0) = 0. x \u2192 0+ x \u2192 0+\u0386\u03c1\u03b1 \u03b7 \u03c8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c4\u03bf 0.\u0388\u03c7\u03bf\u03c5\u03bc\u03b5 \uf8f1 2 1 , x < 0 \uf8f1\uf8f4\uf8f4\u2212 1 , x<0 \uf8f4\uf8f4 2 x > 0 \uf8f2 . \u03c8 \u2032( x) = \uf8f2 \u2212x \u03ba\u03b1\u03b9 \u03c8 \u2032\u2032( x) = \uf8f3\uf8f4\uf8f4\u2212 4x \u2212x \uf8f4 1, 1, x>0 \uf8f3\uf8f4 x 4x x\u03a3\u03c4\u03bf x0 = 0 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 lim \u03c8 (x) \u2212\u03c8 (0) = lim x = lim 1 = +\u221e \u03ba\u03b1\u03b9 x \u2192 0+ x xx \u2192 0+ xx \u2192 0+ lim \u03c8 (x) \u2212\u03c8 (0) = lim \u2212 \u2212x = lim 1 = +\u221e. x \u2192 0\u2212 x xx \u2192 0\u2212 x\u21920\u2212 \u2212x 134","2.8\u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 \u03c8 \u03b4\u03b5\u03bd \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf 0.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 \u03c8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c3\u03c4\u03bf 0, \u03b7 C\u03c8 \u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c4\u03b7\u03c2\u039f(0,0) \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b1\u03ba\u03cc\u03c1\u03c5\u03c6\u03b7 \u03b5\u03c5\u03b8\u03b5\u03af\u03b1 x = 0.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 \u03c8\u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 +\u221e \u03c8\u2033(x) + \u2012 \u03c8(x) 0 \u03a3.\u039a. \u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 \u03c8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03c5\u03c1\u03c4\u03ae \u03c3\u03c4\u03bf (\u2212\u221e, 0] \u03ba\u03b1\u03b9 \u03ba\u03bf\u03af\u03bb\u03b7 \u03c3\u03c4\u03bf [0, +\u221e). \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03c4\u03bf\u03c5 0 \u03b7 \u03c8\u2033 \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03ba\u03b1\u03b9 \u03b7 C\u03c8 \u03b4\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03b5\u03c6\u03b1\u03c0\u03c4\u03bf\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u039f(0,0), \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03b1\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 C\u03c8.4. \u25cf \u0007\u0397 f \u03c3\u03c4\u03bf [\u20131,1] \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03c9\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u2019 \u03b1\u03c5\u03c4\u03cc \u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 (1,1). \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [\u20131,1]. \u039f\u03bc\u03bf\u03af\u03c9\u03c2 \u03b7 f \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [1,4], \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [4,8] \u03ba\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [8,10]. \u25cf \u0397\u0007 f \u03c3\u03c4\u03bf [\u20131,0] \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c7\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03b7 f \u2032 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf (\u20131,0). \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf [\u20131,0]. \u039f\u03bc\u03bf\u03af\u03c9\u03c2 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9\u2014 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf [0,2]\u2014 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf [2,5]\u2014 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf [5,6]\u2014 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf [6,7] \u03ba\u03b1\u03b9\u2014 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03c3\u03c4\u03bf [7,10].\u03a0\u03b9\u03b8\u03b1\u03bd\u03ad\u03c2 \u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ac\u03c4\u03c9\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 1, 4, 6, 8 \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c3\u03c9\u03c4\u03b5\u03c1\u03b9\u03ba\u03ac\u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03c4\u03bf\u03c5 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03b7\u03c2 f \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1 \u03bf\u03c0\u03bf\u03af\u03b1 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b7 f \u2032 , \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b1\u03b9\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u20131, 10 \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03ba\u03c1\u03b1 \u03c4\u03bf\u03c5 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd \u03c4\u03b7\u03c2 f.\u039f\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af 1, 8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03bc\u03b5\u03b3\u03af\u03c3\u03c4\u03c9\u03bd, \u03b5\u03bd\u03ce \u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af \u20131, 4, 10 \u03b5\u03af\u03bd\u03b1\u03b9\u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b5\u03bb\u03b1\u03c7\u03af\u03c3\u03c4\u03c9\u03bd. \u039f \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc\u03c2 6 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b8\u03ad\u03c3\u03b7 \u03c4\u03bf\u03c0\u03b9\u03ba\u03bf\u03cd \u03b1\u03ba\u03c1\u03bf\u03c4\u03ac\u03c4\u03bf\u03c5\u03b1\u03c6\u03bf\u03cd \u03b7 f \u2032 \u03b4\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03bf\u03cd. 135","2.8 \u25cf \u03a4\u03ad\u03bb\u03bf\u03c2, \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 0, 2, 5, 6, 7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03c3\u03b7\u03bc\u03b5\u03af\u03c9\u03bd \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2.5. i) \u0007\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 S \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [0,t2], \u03c4\u03bf \u03ba\u03b9\u03bd\u03b7\u03c4\u03cc \u03b3\u03b9\u03b1 t \u2208[0,t2 ] \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03ba\u03b1\u03c4\u03ac \u03c4\u03b7\u03bd \u03b1\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03ae \u03c6\u03bf\u03c1\u03ac. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 S \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u03c4\u03bf [t2 , +\u221e), \u03c4\u03bf \u03ba\u03b9\u03bd\u03b7\u03c4\u03cc \u03b3\u03b9\u03b1 t \u2265 t2 \u03ba\u03b9\u03bd\u03b5\u03af\u03c4\u03b1\u03b9 \u03ba\u03b1\u03c4\u03ac \u03c4\u03b7 \u03b8\u03b5\u03c4\u03b9\u03ba\u03ae \u03c6\u03bf\u03c1\u03ac. ii) \u0395\u0007 \u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03cc\u03c4\u03b9 \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b9\u03bd\u03b7\u03c4\u03bf\u03cd \u03b5\u03af\u03bd\u03b1\u03b9 \u03c5(t) = S\u2032(t) \u03ba\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03c4\u03b9\u03c2 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ad\u03c2 \u03c3\u03c4\u03b9\u03b3\u03bc\u03ad\u03c2 h\u2032 \u03b7 C \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9 \u03ba\u03b1\u03bc\u03c0\u03ae. \u0391\u03c0\u03cc \u03c4\u03bf \u03c3\u03c7\u03ae\u03bc\u03b1 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9: \u2014 \u03a3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 [0,t1] \u03b7 S \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03ba\u03ac\u03c4\u03c9 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 \u03b7 S\u2032(t) = \u03c5(t) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03c6\u03b8\u03af\u03bd\u03bf\u03c5\u03c3\u03b1 \u03c3\u2019 \u03b1\u03c5\u03c4\u03cc. \u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c3\u03c4\u03bf [0,t1] \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9. \u2014 \u03a3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 [t1,t3] \u03b7 S \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03ac\u03bd\u03c9 \u03ba\u03b1\u03b9 \u03ac\u03c1\u03b1 \u03b7 S\u2032(t) = \u03c5(t) \u03b5\u03af\u03bd\u03b1\u03b9 \u03b3\u03bd\u03b7\u03c3\u03af\u03c9\u03c2 \u03b1\u03cd\u03be\u03bf\u03c5\u03c3\u03b1 \u03c3\u2019 \u03b1\u03c5\u03c4\u03cc. \u0394\u03b7\u03bb\u03b1\u03b4\u03ae \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c3\u03c4\u03bf [t1,t3] \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9. \u2014 \u039f\u03bc\u03bf\u03af\u03c9\u03c2 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c3\u03c4\u03bf [t3, +\u221e) \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9. t0 t1 t3 +\u221e \u03c5(t)\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1 \u03c4\u03bf\u03c5 \u03ba\u03b9\u03bd\u03b7\u03c4\u03bf\u03cd \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03b4\u03b9\u03ac\u03c3\u03c4\u03b7\u03bc\u03b1 [t1,t3] \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1\u03b4\u03b9\u03b1\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 [0,t1] \u03ba\u03b1\u03b9 [t3 , +\u221e) \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9.2.8 \u0392\u0384 \u039f\u039c\u0391\u0394\u0391\u03a31. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: f \u2032( x) = x2 +1\u2212 2x2 = 1\u2212 x2 \u03ba\u03b1\u03b9 (x2 +1)2 (1+ x2 )2 f \u2032(x) = \u22122x(1+ x2 )2 \u2212 2(x2 +1) \u22c5 2x(1\u2212 x2 ) = 2x(x2 \u2212 3) , \u03bf\u03c0\u03cc\u03c4\u03b5 (x2 +1)4 (x2 + 1)3 f \u2032\u2032(x) = 0 \u21d4 x = 0 \u03ae x = \u2212 3 \u03ae x = 3.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. 136","2.8 x \u2212\u221e \u2013 \u22123 + 0 3 +\u221ef \u2033(x) 0 0\u2013 0+f ( x) \u2212 3 0 3 4 \u03a3.\u039a. 4 \u03a3.\u039a. \u03a3.\u039a.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b1 \u2212 3, 0 \u03ba\u03b1\u03b9 3 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03ce\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \uf8eb 3 \uf8f6 \u0393 \uf8eb 3 \uf8f6\u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391 \uf8ed\uf8ec\uf8ec \u2212 3, \u2212 4 \uf8f8\uf8f7\uf8f7 , \u0392(0,0) \u03ba\u03b1\u03b9 \uf8ec\uf8ed\uf8ec 3, 4 \uf8f7\uf8f7\uf8f8 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2\u03c4\u03b7\u03c2 Cf .\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391 \u03ba\u03b1\u03b9 \u0393 \u03ad\u03c7\u03bf\u03c5\u03bd \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c4\u03b5\u03c2 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bc\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03ac\u03c9\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b7\u03bd \u03b1\u03c1\u03c7\u03ae \u03c4\u03c9\u03bd \u03b1\u03be\u03cc\u03bd\u03c9\u03bd \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0392.2. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9:\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(x) = 2ex\u2212\u03b1 \u2212 2x \u03ba\u03b1\u03b9 f \u2032\u2032(x) = 2ex\u2212\u03b1 \u2212 2 = 2(ex\u2212\u03b1 \u22121), f \u2032\u2032(x) = 0 \u21d4 ex\u2212\u03b1 = 1 \u21d4 x = \u03b1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03b7\u03c2 f \u2033 \u03c6\u03b1\u03af\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e \u03b1 +\u221e f \u2033(x) \u20130+ f(x) 2 \u2013 \u03b12 \u03a3.\u039a. \u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03b1 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03bf\u03cd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf, \u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0391(\u03b1 , 2 \u2212\u03b1 2 ), \u03b1 \u2208 R \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cf. \u03a4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03b1\u03c5\u03c4\u03cc \u03b2\u03c1\u03af\u03c3\u03ba\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03b7\u03bd \u03c0\u03b1\u03c1\u03b1\u03b2\u03bf\u03bb\u03ae y = \u2212x2 + 2, \u03b1\u03c6\u03bf\u03cd 2 \u2212 \u03b1 2 = \u2212\u03b1 2 + 2.3. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9: f \u2032(x) = 4x3 \u2212 6\u03b1 x2 +12x + 2 \u03ba\u03b1\u03b9 f \u2032\u2032(x) = 12x2 \u221212\u03b1 x +12 = 12(x2 \u2212\u03b1 x +1). \u03a0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03bf\u03cd\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03b7 f \u2033 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b4\u03b5\u03c5\u03c4\u03b5\u03c1\u03bf\u03b2\u03ac\u03b8\u03bc\u03b9\u03bf \u03c4\u03c1\u03b9\u03ce\u03bd\u03c5\u03bc\u03bf \u03bc\u03b5 \u0394 = \u03b12 \u2013 4 < 0 , \u03b1\u03c6\u03bf\u03cd \u03b1 \u2208 (\u22122, 2). \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2, f \u2033(x) > 0 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R. 137","2.8\u0386\u03c1\u03b1 \u03b7 f \u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c4\u03b1 \u03ba\u03bf\u03af\u03bb\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03b1 \u03ac\u03bd\u03c9 \u03c3\u2019 \u03cc\u03bb\u03bf \u03c4\u03bf R.4. i) \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 x \u2208 R \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 f \u2032(x) = 3x2 \u2212 6x = 3x(x \u2212 2),\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032(x) = 0 \u21d4 x = 0 \u03ae x = 2\u03ba\u03b1\u03b9 f \u2032\u2032(x) = 6x \u2212 6 = 6(x \u22121),\u03bf\u03c0\u03cc\u03c4\u03b5 f \u2032\u2032(x) = 0 \u21d4 x = 1.\u03a4\u03bf \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03c9\u03bd f \u2032 \u03ba\u03b1\u03b9 f \u2033, \u03c4\u03b1 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ac \u03b1\u03ba\u03c1\u03cc\u03c4\u03b1\u03c4\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c6\u03b1\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9\u03c3\u03c4\u03bf\u03bd \u03c0\u03b1\u03c1\u03b1\u03ba\u03ac\u03c4\u03c9 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1. x \u2212\u221e 0 1 2 +\u221ef \u2032 (x) + 0\u2013 \u2013 0+f \u2033(x) \u2013 \u2013 0+ + 2f ( x) T.M. 0 \u20132 \u03a3.\u039a. T.E.\u0394\u03b7\u03bb\u03b1\u03b4\u03ae, \u03b7 f \u03c0\u03b1\u03c1\u03bf\u03c5\u03c3\u03b9\u03ac\u03b6\u03b5\u03b9:\u25cf \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 0 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03bc\u03ad\u03b3\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf f(0) = 2 \u03ba\u03b1\u03b9\u25cf \u03c3\u03c4\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf 2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc \u03b5\u03bb\u03ac\u03c7\u03b9\u03c3\u03c4\u03bf \u03c4\u03bf f(2) = \u2013 2.\u0395\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 f \u2033 \u03bc\u03b7\u03b4\u03b5\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf 1 \u03ba\u03b1\u03b9 \u03b5\u03ba\u03b1\u03c4\u03ad\u03c1\u03c9\u03b8\u03b5\u03bd \u03b1\u03c5\u03c4\u03bf\u03cd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c0\u03c1\u03cc\u03c3\u03b7\u03bc\u03bf \u03c4\u03bf\u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u0393(1,0) \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03bc\u03c0\u03ae\u03c2 \u03c4\u03b7\u03c2 Cf .ii) \u0007\u0393\u03b9\u03b1 \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u0391, \u0392, \u0393 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c5\u03b8\u03b5\u03b9\u03b1\u03ba\u03ac, \u03b1\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03b4\u03b5\u03af\u03be\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03bb\u0391\u0392 = \u03bb\u0391\u0393 . \u0388\u03c7\u03bf\u03c5\u03bc\u03b5: \u03bb\u0391\u0392 = \u22122 \u2212 2 = \u22122 \u03ba\u03b1\u03b9 \u03bb\u0391\u0393 = 0\u22122 = \u22122. 2\u22120 1\u2212 0\u0386\u03c1\u03b1 \u03bb\u0391\u0392 = \u03bb\u0391\u0393 .5. \u0395\u03af\u03bd\u03b1\u03b9: 2 f (x) f \u2032(x) \u2212 2 f \u2032(x) + 2x = 0 138","","","","","","","","","","",""]