{"id":516,"date":"2012-12-10T22:05:31","date_gmt":"2012-12-10T13:05:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=516"},"modified":"2021-09-29T20:09:38","modified_gmt":"2021-09-29T11:09:38","slug":"thr200801","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr200801\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2008\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u591a\u9805\u5f0f \\(f(x)\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u6761\u4ef6 (i) , (ii) , (iii) \u3092\u8003\u3048\u308b.<\/p>\r\n

    \r\n

  1. (i)<\/strong>\u3000\\(x^4 f \\left( \\dfrac{1}{x} \\right) = f(x)\\)<\/p><\/li>\r\n

  2. (ii)<\/strong>\u3000\\(f(1-x) = f(x)\\)<\/p><\/li>\r\n

  3. (iii)<\/strong>\u3000\\(f(1) = 1\\)<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\u6761\u4ef6 (i) \u3092\u307f\u305f\u3059\u591a\u9805\u5f0f \\(f(x)\\) \u306e\u6b21\u6570\u306f \\(4\\) \u4ee5\u4e0b\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\u6761\u4ef6 (i) , (ii) , (iii) \u3092\u3059\u3079\u3066\u307f\u305f\u3059\u591a\u9805\u5f0f \\(f(x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

      \\(f(x) = a _ n x^n + \\cdots +a _ 1 x +a _ 0 \\ ( a _ n \\neq 0 )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nx^4 f \\left( \\dfrac{1}{x} \\right) = a _ 0 x^4 + \\cdots + a _ n x^{4-n}\r\n\\]\r\n\u3053\u308c\u304c\u591a\u9805\u5f0f\u3068\u306a\u308b\u306e\u3067, \u6700\u5f8c\u306e\u9805\u306e\u6b21\u6570\u306b\u7740\u76ee\u3057\u3066\r\n\\[\\begin{align}\r\n4-n & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad n & \\leqq 4\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(f(x)\\) \u306e\u6b21\u6570\u306f \\(4\\) \u4ee5\u4e0b\u3067\u3042\u308b.<\/p>\r\n

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

      (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(f(x) = a _ 4 x^4 +a _ 3 x^3 +a _ 2 x^2 +a _ 1 x +a _ 0\\) \u3068\u304a\u304f.\r\n\\[\r\nx^4 f \\left( \\dfrac{1}{x} \\right) = a _ 0 x^4 +a _ 1 x^3 +a _ 2 x^2 +a _ 3 x +a _ 4\r\n\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6 (i) \u3088\u308a\r\n\\[\r\na _ 3 = a _ 1 , \\ a _ 4 = a _ 0\r\n\\]\r\n\u6761\u4ef6 (ii) , (iii) \u3088\u308a, \\(f(0) = f(1) =1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf(0) & = a _ 0 = 1 , \\\\\r\nf(1) & = 2 +2 a _ 1 +a _ 2 = 1 \\\\\r\n\\text{\u2234} \\quad a _ 2 & = -2 a _ 1 -1 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3055\u3089\u306b, \\(3\\) \u6b21\u306e\u9805\u306b\u7740\u76ee\u3057\u3066\r\n\\[\\begin{align}\r\nf(1-x) & = (1-x)^4 +a _ 1 (1-x)^3 + \\cdots \\\\\r\n& = x^4 +( -a _ 1 -4 ) x^3 + \\cdots\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6 (ii) \u3088\u308a\u4fc2\u6570\u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\\begin{align}\r\n-a _ 1 -4 & = a _ 1 \\\\\r\n\\text{\u2234} \\quad a _ 1 & = -2\r\n\\end{align}\\]\r\n[1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\na _ 2 = 4-1 = 3\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nf(x) = \\underline{x^4 -2x^3 +3x^2 -2x +1}\r\n\\]\r\n\u3053\u308c\u306f\u78ba\u304b\u306b\u6761\u4ef6 (ii) \u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u591a\u9805\u5f0f \\(f(x)\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u6761\u4ef6 (i) , (ii) , (iii) \u3092\u8003\u3048\u308b. (i)\u3000\\(x^4 f \\left( \\dfrac{1}{x} \\right) = f(x)\\) (ii)\u3000\\(f(1-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":[76],"tags":[148,16],"class_list":["post-516","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2008","tag-tohoku_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/516","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=516"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/516\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=516"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=516"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=516"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}