{"id":287,"date":"2012-02-02T20:53:44","date_gmt":"2012-02-02T11:53:44","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=287"},"modified":"2021-09-10T08:45:09","modified_gmt":"2021-09-09T23:45:09","slug":"osr200803","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr200803\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2008\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(N\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u95a2\u6570 \\(f(x) = (N-x) \\log x\\) \u3092 \\(1 \\leqq x \\leqq N\\) \u306e\u7bc4\u56f2\u3067\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u66f2\u7dda \\(y =f(x)\\) \u306f\u4e0a\u306b\u51f8\u3067\u3042\u308a, \u95a2\u6570 \\(f(x)\\) \u306f\u6700\u5927\u5024\u3092 \\(1\\) \u3064\u3060\u3051\u3068\u308b. \u3053\u306e\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u81ea\u7136\u6570\u306e\u5217 \\(a _ 1 , a _ 2 , \\cdots , a _ N\\) \u3092\r\n\\[\r\na _ n = n^{N-n} \\quad ( n =1, 2, \\cdots , N )\r\n\\]\r\n\u3067\u5b9a\u3081\u308b. \\(a _ 1 , a _ 2 , \\cdots , a _ N\\) \u306e\u3046\u3061\u3067\u6700\u5927\u306e\u5024\u3092 \\(M\\) \u3068\u3057, \\(M = a _ n\\) \u3068\u306a\u308b \\(n\\) \u306e\u500b\u6570\u3092 \\(k\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d \\(k \\leqq 2\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u3067 \\(k=2\\) \u3068\u306a\u308b\u306e\u306f, \\(N\\) \u304c \\(2\\) \u306e\u3068\u304d\u3060\u3051\u3067\u3042\u308b\u3053\u3068\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

    \\[\\begin{align}\r\nf'(x) & = -\\log x +(N-x) \\cdot \\dfrac{1}{x} \\\\\r\n& = -\\log x +\\dfrac{N}{x} -1 , \\\\\r\nf''(x) & = -\\dfrac{1}{x} -\\dfrac{N}{x^2} \\\\\r\n& = -\\dfrac{x+N}{x^2} \\geqq 0 \\quad ( \\ \\text{\u2235} \\ 1 \\leqq x \\leqq N \\ )\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(f(x)\\) \u306f \\(1 \\leqq x \\leqq N\\) \u3067\u4e0a\u306b\u51f8\u3067\u3042\u308b.
    \r\n\u307e\u305f, \\(f'(x)\\) \u306f, \u3053\u306e\u533a\u9593\u3067\u5358\u8abf\u6e1b\u5c11\u3057\r\n\\[\r\nf(1) = N-1 \\gt 0 , \\quad f(N) = -\\log N \\lt 0\n\\]\r\n\u306a\u306e\u3067, \\(f'(x) =0\\) \u306f\u89e3 \\(\\alpha \\ ( 1 \\lt \\alpha \\lt N )\\) \u3092 \\(1\\) \u3064\u3060\u3051\u3082\u3064.
    \r\n\u3053\u306e\u3068\u304d, \\(f(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & 1 & \\cdots & \\alpha & \\cdots & N \\\\ \\hline f'(x) & & + & 0 & - & \\\\ \\hline f(x) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(f(x)\\) \u306f \\(1 \\leqq x \\leqq N\\) \u3067 \\(1\\) \u3064\u306e\u6700\u5927\u5024\u3092\u3082\u3064.<\/p>\r\n

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

    \\(\\log a _ n = f(n)\\) \u3067\u3042\u308a, \\(\\log x\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \\(a _ n\\) \u3068 \\(f(n)\\) \u306e\u5897\u6e1b\u306f\u4e00\u81f4\u3059\u308b.
    \r\n\\(x = \\alpha\\) \u306e\u3068\u304d, \\(f(x)\\) \u304c\u6700\u5927\u3068\u306a\u308a, \\(m \\leqq \\alpha \\lt m+1 \\ ( m =1, 2, \\cdots , N-1 )\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308c\u3070, (1)<\/strong> \u306e\u5897\u6e1b\u8868\u3088\u308a, \\(a _ n\\) \u306e\u96a3\u63a5\u3059\u308b\u9805\u306e\u9593\u306b\u306f\r\n\\[\r\na _ 1 \\lt \\cdots \\lt a _ m , \\quad a _ {m+1} \\gt \\cdots \\gt a _ N\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b.
    \r\n\u3088\u3063\u3066, \u6700\u5927\u5024 \\(M\\) \u3092\u53d6\u308a\u3046\u308b\u306e\u306f, \\(a _ {m}\\) \u307e\u305f\u306f \\(a _ {m+1}\\) \u3067\u3042\u308a, \\(k \\leqq 2\\) \u304c\u6210\u7acb\u3059\u308b.<\/p>\r\n

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

    \\(k=2\\) \u3059\u306a\u308f\u3061, \\(a _ {m} =a _ {m+1}\\) \u304c\u6210\u7acb\u3059\u308b\u306e\u306f\r\n\\[\\begin{align}\r\n\\log a _ {m+1} -\\log a _ {m} & = (N-m-1) \\log (m+1) -(N-m) \\log m \\\\\r\n& = (N-m-1) \\log \\dfrac{m+1}{m} +\\log m\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n(N-m-1) \\log \\dfrac{m+1}{m} +\\log m = 0 \\quad ... [1]\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3068\u304d\u3067\u3042\u308b.
    \r\n[1] \u304c\u6210\u7acb\u3059\u308b\u306e\u306f, \\(\\log \\dfrac{m+1}{m} \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nN-m-1 & = 0 \\ \\text{\u304b\u3064} \\ m = 1 \\\\\r\n\\text{\u2234} \\quad N & = 2 , \\ m = 1\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(k=2\\) \u3068\u306a\u308b\u306e\u306f, \\(N=2\\) \u306e\u3068\u304d\u306e\u307f\u3067\u3042\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(N\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. (1)\u3000\u95a2\u6570 \\(f(x) = (N-x) \\log x\\) \u3092 \\(1 \\leqq x \\leqq N\\) \u306e\u7bc4\u56f2\u3067\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u66f2\u7dda \\(y =f(x)\\) … \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":[51],"tags":[142,16],"class_list":["post-287","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2008","tag-osaka_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/287","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=287"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/287\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=287"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=287"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}