{"id":377,"date":"2012-04-21T23:00:30","date_gmt":"2012-04-21T14:00:30","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=377"},"modified":"2021-09-24T17:46:31","modified_gmt":"2021-09-24T08:46:31","slug":"tok201203","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201203\/","title":{"rendered":"\u6771\u5de5\u59272012\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(3\\) \u6b21\u95a2\u6570 \\(y = x^3 -3x^2 +2x\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) , \u76f4\u7dda \\(y = ax\\) \u3092 \\(l\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(C\\) \u3068 \\(l\\) \u304c\u539f\u70b9\u4ee5\u5916\u306e\u5171\u6709\u70b9\u3092\u3082\u3064\u3088\u3046\u306a\u5b9f\u6570 \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u5185\u306b\u3042\u308b\u3068\u304d, \\(C\\) \u3068 \\(l\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S(a)\\) \u3068\u3059\u308b. \\(S(a)\\) \u304c\u6700\u5c0f\u3068\u306a\u308b \\(a\\) \u306e\u5024\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

    \\(C\\) \u3068 \\(l\\) \u306e\u5f0f\u3088\u308a\r\n\\[\\begin{align}\r\nx^3 -3x^2 +2x & = ax \\\\\r\n\\text{\u2234} \\quad x \\left( x^2 -3x +2 -a \\right) & = 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(x\\) \u306e\u65b9\u7a0b\u5f0f \\( x^2 -3x -a+2 = 0 \\quad ... [1]\\) \u304c \\(x = 0\\) \u4ee5\u5916\u306b\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000[1] \u304c \\(x = 0\\) \u3092\u89e3\u306b\u3082\u3064\u3068\u304d\r\n\\[\\begin{align}\r\n-a+2 & = 0 \\\\\r\n\\text{\u2234} \\quad a & = 2\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, [1] \u306f\r\n\\[\\begin{align}\r\nx (x-3) & = 0 \\\\\r\n\\text{\u2234} \\quad x & = 0 , 3\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(x = 0\\) \u4ee5\u5916\u306e\u89e3\u3092\u3082\u3064.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(a \\neq 2\\) \uff08 [1] \u304c \\(x = 0\\) \u3092\u89e3\u306b\u3082\u305f\u306a\u3044\uff09\u3068\u304d
      \r\n[1] \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nD = 3^2 -4(2-a) & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad a & \\geqq -\\dfrac{1}{4}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{a \\geqq -\\dfrac{1}{4}}\r\n\\]\r\n

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

      \\(C\\) \u3068 \\(l\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(R\\) \u3068\u304a\u304f.
      \r\n\\(C\\) \u3068 \\(l\\) \u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(0 , \\alpha , \\beta \\ ( \\alpha \\leqq \\beta )\\) \u3068\u304a\u304f\u3068, [1] \u306b\u3064\u3044\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = 3 , \\ \\alpha \\beta = -a+2 \\quad ... [2]\r\n\\]\r\n[1] \u3092\u89e3\u3051\u3070\r\n\\[\r\nx = \\dfrac{3 \\pm \\sqrt{3^2 -4(2-a)}}{2} = \\dfrac{3 \\pm \\sqrt{4a+1}}{2}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\alpha = \\dfrac{3 -\\sqrt{4a+1}}{2} , \\ \\beta = \\dfrac{3 +\\sqrt{4a+1}}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\dfrac{d \\beta}{da} = -\\dfrac{d \\alpha}{da} \\quad ... [3]\r\n\\]\r\n\u3055\u3089\u306b, \\(f(x) = x^3 -3x^2 +2x\\) \u3068\u304a\u304d, \u539f\u59cb\u95a2\u6570\u306e\u3072\u3068\u3064 \\(F(x)\\) \u3092\r\n\\[\r\nF(x) =\\displaystyle\\int f(x) \\, dx =\\dfrac{x^4}{4} -x^3 +x^2\r\n\\]\r\n\u3068\u304a\u304f.
      \r\n\u3053\u306e\u3068\u304d\r\n\\[\r\nf( \\alpha ) -a \\alpha =0 , \\ f( \\beta ) -a \\beta =0 \\quad ... [4]\r\n\\]\r\n\u4ee5\u4e0b\u3067\u306f, \\(D\\) \u306e\u5f62\u72b6\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(-\\dfrac{1}{4} \\leqq a \\leqq 2\\) \u306e\u3068\u304d
        \r\n\u3053\u306e\u3068\u304d, \\(\\alpha \\beta \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n0 \\leqq \\alpha \\leqq \\beta\r\n\\]\r\n\u3086\u3048\u306b \\(R\\) \u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308a<\/p>\r\n\"\"\r\n

        \\[\\begin{align}\r\nS(a) & = \\displaystyle\\int _ 0^{\\alpha} \\left( f(x) -ax \\right) \\, dx +\\displaystyle\\int _ {\\alpha}^{\\beta} \\left( ax -f(x) \\right) \\, dx \\\\\r\n& = F( \\alpha ) -F(0) -\\dfrac{a \\alpha^2}{2} +\\dfrac{a \\beta^2}{2} -\\dfrac{a \\alpha^2}{2} -F( \\beta ) +F( \\alpha ) \\\\\r\n& = 2F( \\alpha ) -F( \\beta ) -a \\alpha^2 +\\dfrac{a \\beta^2}{2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS'(a) & = 2 f( \\alpha ) \\cdot \\dfrac{d \\alpha}{da} -f( \\beta ) \\cdot \\dfrac{d \\beta}{da} \\\\\r\n& \\qquad -\\alpha^2 -a \\cdot 2\\alpha \\cdot \\dfrac{d \\alpha}{da} +\\dfrac{\\beta^2}{2} +\\dfrac{a}{2} \\cdot 2 \\beta \\cdot \\dfrac{d \\beta}{da} \\\\\r\n& = \\left\\{ 2 \\left( f( \\alpha ) -a \\alpha \\right) +f( \\beta ) -a \\beta \\right\\} \\dfrac{d \\alpha}{da} -\\alpha^2 +\\dfrac{\\beta^2}{2} \\quad ( \\ \\text{\u2235} \\ [3] \\ ) \\\\\r\n& = -\\alpha^2 +\\dfrac{\\beta^2}{2} \\quad ( \\ \\text{\u2235} \\ [4] \\ )\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(\\alpha^2\\) \u306f \\(a\\) \u306e\u5358\u8abf\u6e1b\u5c11\u95a2\u6570, \\(\\beta^2\\) \u306f \\(a\\) \u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\u306a\u306e\u3067, \\(S'(a)\\) \u306f \\(a\\) \u306e\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3067\u3042\u308b.
        \r\n\u306a\u306e\u3067, \\(S'(a) =0\\) \u306f\u9ad8\u3005 \\(1\\) \u3064\u306e\u89e3\u3057\u304b\u3082\u305f\u306a\u3044.
        \r\n\u3053\u308c\u3092\u89e3\u304f\u3068\r\n\\[\\begin{align}\r\n\\alpha^2 & = \\dfrac{\\beta^2}{2} \\\\\r\n\\text{\u2234} \\quad \\beta & = \\sqrt{2} \\alpha\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [2] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\alpha & = \\dfrac{3}{\\sqrt{2} +1} =3 \\left( \\sqrt{2} -1 \\right) , \\\\\r\na & = 2 -\\sqrt{2} \\alpha^2 =2 -9 \\sqrt{2} \\left( \\sqrt{2} -1 \\right)^2 \\\\\r\n& = 38 -27 \\sqrt{2} = -0.8 \\cdots\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(S(a)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} a & -\\frac{1}{4}& \\cdots & 38 -27 \\sqrt{2} & \\cdots & 2 \\\\ \\hline S'(a) & & - & 0 & + & \\\\ \\hline S(a) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & \\end{array}\r\n\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(a \\gt 2\\) \u306e\u3068\u304d
        \r\n\u3053\u306e\u3068\u304d, \\(\\alpha \\beta \\lt 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\alpha \\lt 0 \\lt \\beta\r\n\\]\r\n\u3086\u3048\u306b \\(R\\) \u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

        \u3053\u306e\u3068\u304d, \u660e\u3089\u304b\u306b \\(S(a) \\gt S(2)\\) \u306a\u306e\u3067, \u3053\u306e\u7bc4\u56f2\u3067\u6700\u5c0f\u5024\u3092\u3068\u308b\u3053\u3068\u306f\u306a\u3044.<\/p><\/li>\r\n<\/ol>\r\n

        1*<\/strong> 2*<\/strong>\u3088\u308a, \\(S(a)\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na =\\underline{38 -27 \\sqrt{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(3\\) \u6b21\u95a2\u6570 \\(y = x^3 -3x^2 +2x\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) , \u76f4\u7dda \\(y = ax\\) \u3092 \\(l\\) \u3068\u3059\u308b. (1)\u3000\\(C\\) \u3068 \\(l\\) \u304c\u539f\u70b9\u4ee5\u5916\u306e\u5171\u6709\u70b9\u3092\u3082\u3064\u3088\u3046\u306a\u5b9f\u6570 \\ … \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":[61],"tags":[141,68],"class_list":["post-377","post","type-post","status-publish","format-standard","hentry","category-toko_2012","tag-toko","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/377","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=377"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/377\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=377"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=377"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}