\\(f(x) = x^3+3x^2-9x\\) \u3068\u3059\u308b.\r\n\\(y \\lt x \\lt a\\) \u3092\u6e80\u305f\u3059\u3059\u3079\u3066\u306e \\(x , y\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nf(x) \\gt \\dfrac{(x-y) f(a) +(a-x) f(y)}{a-y}\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\[\\begin{align}\r\nf(x) & \\gt \\dfrac{(x-y) f(a) +(a-x) f(y)}{a-y} \\\\\r\n(a-y) f(x) +x f(x) & \\gt (x-y) f(a) +(a-x) f(y) +x f(x) \\\\\r\n(x-y) \\{ f(x) -f(a) \\} & \\gt (a-x) \\{ f(y) -f(x) \\} \\\\\r\n\\text{\u2234} \\quad \\dfrac{f(x) -f(y)}{x-y} & \\gt \\dfrac{f(a) -f(x)}{a-x} \\quad ... [1]\r\n\\end{align}\\]\r\n\u4e2d\u9593\u5024\u306e\u5b9a\u7406\u3088\u308a, \\(y \\lt c _ 1 \\lt x \\lt c _ 2 \\lt a\\) \u3067\r\n\\[\r\nf'(c _ 1) = \\dfrac{f(x) -f(y)}{x-y} , \\ f'(c _ 2) = \\dfrac{f(a) -f(x)}{a-x}\r\n\\]\r\n\u3092\u307f\u305f\u3059 \\(c _ 1 , c _ 2\\) \u304c\u5b58\u5728\u3059\u308b.
\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, [1] \u306f\r\n\\[\r\nf'(c _ 1) \\gt f'(c _ 2)\r\n\\]\r\n\u3053\u308c\u304c\u5e38\u306b\u6210\u7acb\u3059\u308b\u306e\u306f, \\(x \\leqq a\\) \u306b\u304a\u3044\u3066, \\(f'(x)\\) \u304c\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u3068\u304d, \u3059\u306a\u308f\u3061 \\(f''(x) \\lt 0\\) \u3068\u306a\u308b\u3068\u304d\u3067\u3042\u308b.\r\n\\[\\begin{align}\r\nf'(x) & = 3x^2 +6x , \\\\\r\nf''(x) & = 6x +6 =6(x+1)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nf''(a) & = 6(a+1) \\leqq 0 \\\\\r\n\\text{\u2234} \\quad & \\underline{a \\leqq -1}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = x^3+3x^2-9x\\) \u3068\u3059\u308b. \\(y \\lt x \\lt a\\) \u3092\u6e80\u305f\u3059\u3059\u3079\u3066\u306e \\(x , y\\) \u306b\u5bfe\u3057\u3066 \\[ f(x) \\gt \\dfrac{(x-y) f(a) +(a-x) f(y … \u7d9a\u304d\u3092\u8aad\u3080