{"id":116,"date":"2011-11-28T00:07:15","date_gmt":"2011-11-27T15:07:15","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=116"},"modified":"2021-09-09T21:07:41","modified_gmt":"2021-09-09T12:07:41","slug":"osr201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201001\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\u95a2\u6570\r\n\\[\r\nf(x) = 2 \\log \\left( 1+e^x \\right) -x -\\log 2\r\n\\]\r\n\u3092\u8003\u3048\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(x)\\) \u306e\u7b2c \\(2\\) \u6b21\u5c0e\u95a2\u6570\u3092 \\(f''(x)\\) \u3068\u3059\u308b. \u7b49\u5f0f\r\n\\[\r\n\\log f''(x) = -f(x)\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 0^{\\log 2} \\left( x-\\log 2 \\right) e^{-f(x)} \\, dx\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nf'(x) & = 2 \\cdot \\dfrac{e^x}{1+e^x} -1 \\\\\r\n& = 2 -\\dfrac{2}{1+e^x} -1 \\\\\r\n& = 1 -\\dfrac{2}{1+e^x} , \\\\\r\nf''(x) & = \\dfrac{2 e^x}{\\left( 1+e^x \\right)^2}\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\log f''(x) & = \\log 2 +\\log e^x -2 \\log \\left( 1+e^x \\right) \\\\\r\n& = -2 \\log \\left( 1+e^x \\right) +x +\\log 2 \\\\\r\n& = -f(x)\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf''(x) = e^{-f(x)}\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^{\\log 2} & \\left( x-\\log 2 \\right) e^{-f(x)} \\, dx = \\displaystyle\\int _ 0^{\\log 2} \\left( x-\\log 2 \\right) f''(x) \\, dx \\\\\r\n& = \\left[ \\left( x-\\log 2 \\right) f'(x) \\right] _ 0^{\\log 2} - \\int _ 0^{\\log 2} 1 \\cdot f'(x) \\, dx \\\\\r\n& = 0 \\cdot f'( \\log 2 ) -\\left( -\\log 2 \\right) f'(0) -\\left[ f(x) \\right] _ 0^{\\log 2} \\\\\r\n& = \\log 2 \\cdot 0 - \\left[ f(x) \\right] _ 0^{\\log 2} \\quad ( \\ \\text{\u2235} \\ f'(0) = 1 -\\dfrac{2}{1+e^0} = 0 \\ ) \\\\\r\n& = f(0) -f( \\log 2 ) \\\\\r\n& = \\left( 2 \\log 2 -\\log 2 \\right) - \\left\\{ 2 \\log (1+2) -\\log 2 -\\log 2 \\right\\} \\\\\r\n& = \\underline{3 \\log 2 - 2 \\log 3}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\[ f(x) = 2 \\log \\left( 1+e^x \\right) -x -\\log 2 \\] \u3092\u8003\u3048\u308b. (1)\u3000\\(f(x)\\) \u306e\u7b2c \\(2\\) \u6b21\u5c0e\u95a2\u6570\u3092 \\(f''(x)\\) \u3068\u3059\u308b. \u7b49\u5f0f \\[ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[31],"tags":[142,14],"class_list":["post-116","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2010","tag-osaka_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/116\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}