{"id":1121,"date":"2015-06-21T23:03:10","date_gmt":"2015-06-21T14:03:10","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1121"},"modified":"2021-09-09T06:33:03","modified_gmt":"2021-09-08T21:33:03","slug":"osr201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201402\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2014\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(t \\gt 0\\) \u306b\u304a\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(t)\\) \u306f\u6b21\u306e\u6761\u4ef6 (\u30a2) (\u30a4) \u3092\u6e80\u305f\u3059.<\/p>\r\n<ol>\r\n<li><p>(\u30a2)\u3000\\(t \\gt 0\\) \u306e\u3068\u304d, \u3059\u3079\u3066\u306e\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066\u4e0d\u7b49\u5f0f\r\n\\[\r\nt \\cdot \\dfrac{e^x +e^{-x}}{2} +f(t) \\geqq 1+x\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064.<\/p><\/li>\r\n<li><p>(\u30a4)\u3000\\(t \\gt 0\\) \u306b\u5bfe\u3057\u3066, \u7b49\u5f0f\r\n\\[\r\nt \\cdot \\dfrac{e^x +e^{-x}}{2} +f(t) = 1+x\r\n\\]\r\n\u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(x\\) \u304c\u5b58\u5728\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\u3053\u306e\u3068\u304d, \\(f(t)\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p>\\(g(x) = t \\cdot \\dfrac{e^x +e^{-x}}{2} +f(t) -(1+x)\\) \u3068\u304a\u304f.\r\n\\[\r\ng'(x) = t \\cdot \\dfrac{e^x -e^{-x}}{2} -1 \\ .\r\n\\]\r\n\\(g'(x) = 0\\) \u3092\u3068\u304f\u3068, \\(t \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ne^x -e^{-x} & = \\dfrac{2}{t} \\\\\r\nt e^{2x} -2 e^x -t & = 0 \\\\\r\n\\text{\u2234} \\quad e^x = \\dfrac{1 +\\sqrt{t^2 +1}}{t} & \\quad ( \\ \\text{\u2235} \\quad e^x \\gt 0 \\ ) \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d\u306e \\(x\\) \u306e\u5024\u3092 \\(\\alpha \\left( = \\log \\left( 1 +\\sqrt{t^2 +1} \\right) -\\log t \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ne^{-\\alpha} = e^{\\alpha} -\\dfrac{t}{2} = \\dfrac{-1 +\\sqrt{t^2 +1}}{t} \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066<\/p>\r\n<ul>\r\n<li><p>\\(x \\lt \\alpha\\) \u306e\u3068\u304d, \\(g'(x) \\lt 0\\)<\/p><\/li>\r\n<li><p>\\(x \\gt \\alpha\\) \u306e\u3068\u304d, \\(g'(x) \\gt 0\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3086\u3048\u306b, \\(g(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccc} x & \\cdots & \\alpha & \\cdots \\\\ \\hline g'(x) & - & 0 & + \\\\ \\hline g(x) & \\searrow & \\text{\u6700\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6761\u4ef6 (\u30a2) (\u30a4) \u3092\u307f\u305f\u3059\u305f\u3081\u306e\u6761\u4ef6\u306f\r\n\\[\r\ng( \\alpha ) = 0 \\ .\r\n\\]\r\n\u3053\u308c\u3092\u3068\u304f\u3068\r\n\\[\r\nt \\cdot \\dfrac{2 \\sqrt{t^2 +1}}{2t} +f(t) -\\left( 1 +\\log \\left( 1 +\\sqrt{t^2 +1} \\right) -\\log t \\right) = 0 \\\\\r\n\\text{\u2234} \\quad f(t) = \\underline{\\log \\left( 1 +\\sqrt{t^2 +1} \\right) -\\log t -\\sqrt{t^2 +1} +1} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(t \\gt 0\\) \u306b\u304a\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570 \\(f(t)\\) \u306f\u6b21\u306e\u6761\u4ef6 (\u30a2) (\u30a4) \u3092\u6e80\u305f\u3059. (\u30a2)\u3000\\(t \\gt 0\\) \u306e\u3068\u304d, \u3059\u3079\u3066\u306e\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066\u4e0d\u7b49\u5f0f \\[ t \\cdot \\dfr &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/osr201402\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[115],"tags":[142,112],"class_list":["post-1121","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2014","tag-osaka_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1121","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=1121"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1121\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}