{"id":459,"date":"2012-11-01T23:19:34","date_gmt":"2012-11-01T14:19:34","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=459"},"modified":"2021-10-03T15:58:04","modified_gmt":"2021-10-03T06:58:04","slug":"kbr201003","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201003\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2010\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \\(1\\) \u304b\u3089 \\(n\\) \u307e\u3067\u306e\u81ea\u7136\u6570\u306e\u7a4d\u3092 \\(n !\\) \u3067\u8868\u3059.\r\n\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5358\u8abf\u306b\u5897\u52a0\u3059\u308b\u9023\u7d9a\u95a2\u6570 \\(f(x)\\) \u306b\u5bfe\u3057\u3066, \u4e0d\u7b49\u5f0f \\(\\displaystyle\\int _ {k-1}^k f(x) \\, dx \\leqq f(k)\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4e0d\u7b49\u5f0f \\(\\displaystyle\\int _ 1^n \\log x \\, dx \\leqq \\log n !\\) \u3092\u793a\u3057, \u4e0d\u7b49\u5f0f \\(n^n e^{1-n} \\leqq n !\\) \u3092\u5c0e\u3051.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(x \\geqq 0\\) \u306b\u5bfe\u3057\u3066, \u4e0d\u7b49\u5f0f \\(x^n e^{1-n} \\leqq n !\\) \u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u95a2\u6570\u306a\u306e\u3067, \\(k-1 \\leqq x \\leqq k\\) \u306b\u304a\u3044\u3066\r\n\\[\r\nf(x) \\leqq f(k)\r\n\\]\r\n\u4e21\u8fba\u3092 \\(x\\) \u306b\u3064\u3044\u3066 \\(k-1 \\rightarrow k\\) \u307e\u3067\u7a4d\u5206\u3059\u308c\u3070\r\n\\[\r\n\\displaystyle\\int _ {k-1}^k f(x) \\, dx \\leqq \\displaystyle\\int _ {k-1}^k f(k) \\, dx = f(k)\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\int _ {k-1}^k f(x) \\, dx \\leqq f(k)\r\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u306b\u5bfe\u3057\u3066, \\(f(x) = \\log x\\) \uff08\u5358\u8abf\u5897\u52a0\u95a2\u6570\uff09\u3068\u304a\u304d, \\(k = 2, \\cdots , n\\) \u3092\u4ee3\u5165\u3057\u305f\u8fba\u3005\u3092\u52a0\u3048\u308b\u3068\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 1^2 \\log x \\, dx + \\cdots +\\displaystyle\\int _ {n-1}^n \\log x \\, dx & \\leqq \\log 2 + \\cdots + \\log n \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\int _ 1^n \\log x \\, dx & \\leqq \\log n ! \\quad ( \\ \\text{\u2235} \\ \\log 1 = 0\\ )\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 1^n \\log x \\, dx & = \\left[ x \\log x \\right] _ 1^n -\\displaystyle\\int _ 1^n x \\cdot \\dfrac{1}{x} \\, dx \\\\\r\n& = n \\log n +1-n \\\\\r\n& = \\log n^n e^{1-n}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u771f\u6570\u3092\u3068\u308c\u3070\r\n\\[\r\nn^n e^{1-n} \\leqq n !\r\n\\]\r\n

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

    \\(g(x) =x^n e^{1-n}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\ng'(x) & = nx^{n-1} e^{1-n} -x^n e^{1-n} \\\\\r\n& = (n-x) x^{n-1} e^{1-n}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(x \\geqq 0\\) \u306b\u304a\u3051\u308b \\(g(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} x & 0 & \\cdots & n & \\cdots \\\\ \\hline f'(x) & & + & 0 & - \\\\ \\hline f(x) & & \\nearrow & \\text{\u6700\u5927} & \\searrow \\end{array}\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\ng(x) \\leqq g(n) = n^n e^{1-n}\r\n\\]\r\n\u3088\u3063\u3066, \u3053\u308c\u3068 (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nx^n e^{1-x} \\leqq n !\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \\(1\\) \u304b\u3089 \\(n\\) \u307e\u3067\u306e\u81ea\u7136\u6570\u306e\u7a4d\u3092 \\(n !\\) \u3067\u8868\u3059. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u5358\u8abf\u306b\u5897\u52a0\u3059\u308b\u9023\u7d9a\u95a2\u6570 \\(f(x)\\) \u306b\u5bfe\u3057\u3066, \u4e0d\u7b49\u5f0f \\(\\displ … \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":[40],"tags":[144,68],"class_list":["post-459","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2010","tag-tsukuba_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/459","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=459"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/459\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=459"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=459"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=459"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}