\\(0 \\leqq \\alpha \\leqq \\beta\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(\\alpha , \\beta\\) \u3068, \\(2\\) \u6b21\u5f0f \\(f(x) = x^2 -( \\alpha +\\beta )x + \\alpha \\beta\\) \u306b\u3064\u3044\u3066,\r\n\\[\r\n\\displaystyle\\int _ {-1}^1 f(x) \\, dx = 1\r\n\\]\r\n\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u5b9a\u7a4d\u5206\r\n\\[\r\nS = \\displaystyle\\int _ {0}^{\\alpha} f(x) \\, dx\r\n\\]\r\n\u3092 \\(\\alpha\\) \u306e\u5f0f\u3067\u8868\u3057, \\(S\\) \u304c\u3068\u308a\u3046\u308b\u5024\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\[\\begin{align}\r\n\\displaystyle\\int _ {-1}^1 f(x) \\, dx & = 2 \\displaystyle\\int _ {0}^1 ( x^2 +\\alpha \\beta ) \\, dx \\\\\r\n& = 2 \\left[ \\dfrac{x^3}{3} +\\alpha \\beta x \\right] _ 0^1 \\\\\r\n& = 2 \\left( \\dfrac{1}{3} +\\alpha \\beta \\right)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n2 \\left( \\dfrac{1}{3} +\\alpha \\beta \\right) = 1 \\\\\r\n\\text{\u2234} \\quad \\alpha \\beta = \\dfrac{1}{6}\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\alpha \\neq 0\\) \u306e\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u304f\r\n\\[\r\n\\beta = \\dfrac{1}{6 \\alpha}\r\n\\]\r\n\\(0 \\lt \\alpha \\leqq \\beta\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\alpha & \\leqq \\dfrac{1}{6 \\alpha} \\\\\r\n\\text{\u2234} \\quad 0 & \\lt \\alpha \\leqq \\dfrac{1}{\\sqrt{6}} \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nS & = \\left[ \\dfrac{x^3}{3} -\\dfrac{\\alpha +\\beta}{2} x^2 +\\alpha \\beta x \\right] _ 0^{\\alpha} \\\\\r\n& = \\dfrac{\\alpha^3}{3} -\\dfrac{\\alpha^2}{2} \\left( \\alpha +\\dfrac{1}{6 \\alpha} \\right) +\\dfrac{\\alpha}{6} \\\\\r\n& = \\underline{\\dfrac{1}{12} ( \\alpha -2 \\alpha^3 )}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{d S}{d \\alpha} & = \\dfrac{1}{12} ( 1 -6 \\alpha^2 ) \\\\\r\n& = -\\dfrac{1}{12} \\left( \\alpha +\\dfrac{1}{\\sqrt{6}} \\right) \\left( \\alpha -\\dfrac{1}{\\sqrt{6}} \\right) \\geqq 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(\\alpha = \\dfrac{1}{\\sqrt{6}}\\) \u306e\u3068\u304d, \\(S\\) \u306f\u6700\u5927\u5024\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{12} \\left( \\dfrac{1}{\\sqrt{6}} -\\dfrac{2}{6 \\sqrt{6}} \\right) \\\\\r\n& = \\dfrac{1}{12} \\cdot \\dfrac{2}{3 \\sqrt{6}} \\\\\r\n& = \\underline{\\dfrac{\\sqrt{6}}{108}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\leqq \\alpha \\leqq \\beta\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(\\alpha , \\beta\\) \u3068, \\(2\\) \u6b21\u5f0f \\(f(x) = x^2 -( \\alpha +\\beta )x + \\alpha … \u7d9a\u304d\u3092\u8aad\u3080