\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(0 \\lt x \\lt \\pi\\) \u306e\u3068\u304d,\r\n\\[\r\n\\sin x -x \\cos x \\gt 0\r\n\\]\r\n\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u5b9a\u7a4d\u5206\r\n\\[\r\nI = \\displaystyle\\int _ 0^\\pi \\left| \\sin x - ax \\right| \\, dx \\quad ( 0 \\lt a \\lt 1 )\r\n\\]\r\n\u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(f(x) = \\sin x -x \\cos x\\) \u3068\u304a\u304f.\r\n\\[\r\nf'(x) = \\cos x -\\cos x +x \\sin x = x \\sin x\r\n\\]\r\n\\(0 \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066 \\(f'(x) \\gt 0\\) \u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3059\u308b\u304b\u3089\r\n\\[\r\nf(x) \\gt f(0) = 0\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\sin x -x \\cos x \\gt 0\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(0 \\lt a \\lt 1\\) \u306e\u3068\u304d, \\(0 \\lt x \\lt \\pi\\) \u306b\u304a\u3044\u3066 \\(y = \\sin x\\) \u3068 \\(y = ax\\) \u306f \\(1\\) \u3064\u306e\u4ea4\u70b9\u3092\u3082\u3064.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u3053\u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\n\\sin p & = ap \\\\\r\n\\text{\u2234} \\quad a & = \\dfrac{\\sin p}{p} \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nI & = \\displaystyle\\int _ 0^p ( \\sin x -ax ) \\, dx + \\displaystyle\\int _ p^{\\pi} ( ax -\\sin x ) \\, dx \\\\\r\n& = \\left[ -\\cos x -\\dfrac{ax^2}{2} \\right] _ 0^p + \\left[ \\cos x +\\dfrac{ax^2}{2} \\right] _ p^{\\pi} \\\\\r\n& = -2 \\left( \\cos p +\\dfrac{ap^2}{2} \\right) +1 -1 +\\dfrac{a {\\pi}^2}{2} \\\\\r\n& = -2 \\cos p -p \\sin p +\\dfrac{{\\pi}^2}{2} \\cdot \\dfrac{\\sin p}{p}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\n\\dfrac{dI}{dp} & = 2 \\sin p -\\sin p -p \\cos p +\\dfrac{{\\pi}^2}{2} \\cdot \\dfrac{p \\cos p -\\sin p}{p^2} \\\\\r\n& = \\left( 1 -\\dfrac{{\\pi}^2}{2p} \\right) ( \\sin p -p \\cos p )\r\n\\end{align}\\]\r\n\\(0 \\lt p \\lt \\pi\\) \u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\sin p -p \\cos p \\gt 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\dfrac{dI}{dp}= 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\\begin{align}\r\n\\dfrac{{\\pi}^2}{2p^2} & = 1 \\\\\r\n\\text{\u2234} \\quad p & = \\dfrac{\\sqrt{2} \\pi}{2}\r\n\\end{align}\\]\r\n\\(I\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} p & 0 & \\cdots & \\dfrac{\\sqrt{2} \\pi}{2} & \\cdots & \\pi \\\\ \\hline \\frac{dI}{dp} & & - & 0 & + & \\\\ \\hline I & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & \\\\ \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \\(p = \\dfrac{\\sqrt{2} \\pi}{2}\\) \u306e\u3068\u304d \\(I\\) \u306f\u6700\u5c0f\u3068\u306a\u308a, \u6c42\u3081\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na = \\underline{\\dfrac{\\sqrt{2}}{\\pi} \\sin \\dfrac{\\sqrt{2} \\pi}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(0 \\lt x \\lt \\pi\\) \u306e\u3068\u304d, \\[ \\sin x -x \\cos x \\gt 0 \\] \u3092\u793a\u305b. (2)\u3000\u5b9a\u7a4d\u5206 \\[ I = \\displaystyle\\int _ 0^ … \u7d9a\u304d\u3092\u8aad\u3080