\\(f(x) = \\log \\left( e^x +e^{-x} \\right)\\) \u3068\u304a\u304f.\r\n\u66f2\u7dda \\(y = f(x)\\) \u306e\u70b9 \\(( t , f(t) )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b.\r\n\u76f4\u7dda \\(\\ell\\) \u3068 \\(y\\) \u8ef8\u306e\u4ea4\u70b9\u306e \\(y\\) \u5ea7\u6a19\u3092 \\(b(t)\\) \u3068\u304a\u304f.<\/p>\r\n
(1)<\/strong>\u3000\u6b21\u306e\u7b49\u5f0f\u3092\u793a\u305b.\r\n\\[\r\nb(t) = \\dfrac{2t e^{-t}}{e^t +e^{-t}} +\\log \\left( 1 +e^{-2t} \\right)\r\n\\]<\/li>\r\n (2)<\/strong>\u3000\\(x \\geqq 0\\) \u306e\u3068\u304d, \\(\\log (1+x) \\leqq x\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(t \\geqq 0\\) \u306e\u3068\u304d,\r\n\\[\r\nb(t) \\leqq \\dfrac{2}{e^t +e^{-t}} +e^{-2t}\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (4)<\/strong>\u3000\\(b(0) = \\displaystyle\\lim _ {x \\rightarrow \\infty} \\displaystyle\\int _ 0^x \\dfrac{4t}{\\left( e^t +e^{-t} \\right)^2} \\, dt\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf'(x) & = \\dfrac{e^x -e^{-x}}{e^x +e^{-x}} \\\\\r\n& = 1 -\\dfrac{2 e^{-x}}{e^x +e^{-x}} \\quad ... [1]\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(\\ell\\) \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = (x-t) f'(t) +f(t) \\\\\r\n& = x f'(t) +f(t) -t f'(t)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nb(t) & = f(t) -t f'(t) \\quad ... [2] \\\\\r\n& = t +\\log ( 1 +e^{-2t} ) -t +\\dfrac{2 e^{-2t}}{e^t +e^{-t}} \\\\\r\n& = \\dfrac{2t e^{-t}}{e^t +e^{-t}} +\\log \\left( 1 +e^{-2t} \\right)\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(g(x) = x -\\log (1+x)\\) \u3068\u304a\u304f\u3068, \\(x \\geqq 0\\) \u306b\u304a\u3044\u3066\r\n\\[\r\ng'(x) = 1 -\\dfrac{1}{1+x} = \\dfrac{x}{1+x} \\geqq 0\r\n\\]\r\n\u3086\u3048\u306b, \\(x \\geqq 0\\) \u3067 \\(g(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057\r\n\\[\r\ng(x) \\geqq g(0) = 0\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\log (1+x) \\leqq x\r\n\\]\r\n (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(x \\geqq 0\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nx & \\leqq 1+x \\leqq e^x \\\\\r\n\\text{\u2234} \\quad & x e^{-x} \\leqq 1\r\n\\end{align}\\]\r\n\u3053\u308c\u3068, (2)<\/strong> \u306e\u7d50\u679c\u3092 \\(x \\rightarrow e^{-2t}\\) \u306b\u7f6e\u63db\u3048\u305f\u5f0f\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\nb(t) \\leqq \\dfrac{2}{e^t +e^{-t}} +e^{-2t}\r\n\\]\r\n (4)<\/strong><\/p>\r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\nf''(x) & = -2 \\cdot \\dfrac{-e^{-x} ( e^x +e^{-x} ) -e^{-x} ( e^x -e^{-x} )}{( e^x +e^{-x} )^2} \\\\\r\n& = \\dfrac{4}{e^x +e^{-x}}\r\n\\end{align}\\]\r\n\u307e\u305f, [2] \u304b\u3089\r\n\\[\\begin{align}\r\nb'(t) & = -f'(t) -t f''(t) +f'(t) \\\\\r\n& = -t f''(t) = -\\dfrac{4t}{( e^x +e^{-x} )^2}\r\n\\end{align}\\]\r\n\u5fae\u5206\u3068\u7a4d\u5206\u306e\u95a2\u4fc2\u304b\u3089\r\n\\[\\begin{align}\r\nb(x) -b(0) & = \\displaystyle\\int _ 0^x b'(t) \\, dt = -\\displaystyle\\int _ 0^x \\dfrac{4t}{( e^x +e^{-x} )^2} \\, dt \\\\\r\n\\text{\u2234} \\quad & b(0) -b(x) = \\displaystyle\\int _ 0^x \\dfrac{4t}{( e^x +e^{-x} )^2} \\, dt \\quad ... [3]\r\n\\end{align}\\]\r\n(3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n0 \\leqq b(x) \\leqq \\underline{\\dfrac{2}{e^x +e^{-x}} +e^{-2x}} _ {[4]}\r\n\\]\r\n\u3053\u3053\u3067, \\(\\displaystyle\\lim _ {x \\rightarrow \\infty} [4] = 0\\) \u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow \\infty} b(x) = 0\r\n\\]\r\n\u3088\u3063\u3066, [3] \u306b\u3064\u3044\u3066\u3082 \\(x \\rightarrow \\infty\\) \u306e\u3068\u304d\u3092\u8003\u3048\u308c\u3070\r\n\\[\r\nb(0) = \\displaystyle\\lim _ {x \\rightarrow \\infty} \\displaystyle\\int _ 0^x \\dfrac{4t}{( e^x +e^{-x} )^2} \\, dt\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = \\log \\left( e^x +e^{-x} \\right)\\) \u3068\u304a\u304f. \u66f2\u7dda \\(y = f(x)\\) \u306e\u70b9 \\(( t , f(t) )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(\\ell\\) \u3068\u3059\u308b. \u76f4\u7dda \\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n