{"id":1285,"date":"2015-09-23T00:12:54","date_gmt":"2015-09-22T15:12:54","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1285"},"modified":"2021-09-16T09:45:16","modified_gmt":"2021-09-16T00:45:16","slug":"thr201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201502\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \\(3\\) \u6b21\u95a2\u6570 \\(y = x^3 -x\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) \u3068\u3057, \u4e0d\u7b49\u5f0f\r\n\\[\r\nx^3 -x \\gt y \\gt -x\r\n\\]\r\n\u306e\u8868\u3059\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b. \u307e\u305f, P \u3092 \\(D\\) \u306e\u70b9\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000P \u3092\u901a\u308a, \\(C\\) \u306b\u63a5\u3059\u308b\u76f4\u7dda\u304c \\(3\\) \u672c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000P \u3092\u901a\u308a, \\(C\\) \u306b\u63a5\u3059\u308b \\(3\\) \u672c\u306e\u76f4\u7dda\u306e\u50be\u304d\u306e\u548c\u3068\u7a4d\u304c\u3068\u3082\u306b \\(0\\) \u306b\u306a\u308b\u3088\u3046\u306a P \u306e\u5ea7\u6a19\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

    P \\(( p , q )\\) \u3068\u304a\u304f\u3068, \u6761\u4ef6\u304b\u3089\r\n\\[\r\np \\gt 0 , \\ 0 \\lt p+q \\lt p^3 \\quad ... [1] \\ .\r\n\\]\r\n\\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = 3x^2 -1 \\ .\r\n\\]\r\n\\(C\\) \u4e0a\u306e\u70b9 \\(( t , t^3 -t )\\) \u306b\u304a\u3051\u308b \\(C\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = ( 3t^2 -1 ) (x-t) +t^3 -t \\\\\r\n& = ( 3t^2 -1 ) x -2t^3 \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u304c P \u3092\u901a\u308b\u306a\u3089\u3070\r\n\\[\\begin{gather}\r\nq = ( 3t^2 -1 ) p -2t^3 \\\\\r\n\\text{\u2234} \\quad 2t^3 -3p t^2 +p +q = 0 \\quad ... [2] \\ .\r\n\\end{gather}\\]\r\n\\(t\\) \u306e\u65b9\u7a0b\u5f0f [2] \u304c \\(3\\) \u3064\u306e\u7570\u306a\u308b\u5b9f\u6570\u89e3\u3092\u3082\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
    \r\n[2] \u306e\u5de6\u8fba\u3092 \\(f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(t) = 6t^2 -6pt = 6t (t-x) \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & \\cdots & 0 & \\cdots & p & \\cdots \\\\ \\hline f'(t) & + & 0 & - & 0 & + \\\\ \\hline f(t) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3053\u3053\u3067, [1] \u3088\u308a\r\n\\[\\begin{align}\r\nf(0) & = p+q \\gt 0 \\ , \\\\\r\nf(p) & = -p^3 +p +q \\lt 0 \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(y = f(t)\\) \u306e\u30b0\u30e9\u30d5\u3068 \\(y = 0\\) \u306f\u7570\u306a\u308b \\(3\\) \u3064\u306e\u4ea4\u70b9\u3092\u3082\u3064.
    \r\n\u3088\u3063\u3066, \u65b9\u7a0b\u5f0f [2] \u306f\u7570\u306a\u308b \\(3\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3061, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

    (2)<\/strong><\/p>\r\n[2] \u306e \\(3\\) \u3064\u306e\u89e3\u3092 \\(\\alpha , \\beta \\gamma\\) \u3068\u304a\u304f\u3068, \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\left\\{ \\begin{array}{ll} 3 ( {\\alpha}^2 +{\\beta}^2 +{\\gamma}^2) -3 = 0 & ... [2] \\\\ ( 3 {\\alpha}^2 -1 ) ( 3 {\\beta}^2 -1 ) ( 3 {\\gamma}^2 -1 ) = 0 & ... [3] \\end{array} \\right. \\ .\r\n\\]\r\n\u307e\u305f, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\\begin{align}\r\n\\alpha +\\beta +\\gamma & = \\dfrac{3p}{2} \\\\\r\n\\alpha \\beta +\\beta \\gamma +\\gamma \\alpha & = 0 \\\\\r\n\\alpha \\beta \\gamma & = -\\dfrac{p+q}{2} \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3068, [2] \u3088\u308a\r\n\\[\\begin{align}\r\n( \\alpha +\\beta +\\gamma )^2 -2 ( \\alpha \\beta & +\\beta \\gamma +\\gamma \\alpha ) -1 = 0 \\\\\r\n\\dfrac{9p^2}{4} & = 1\r\n\\\\ \\text{\u2234} \\quad p & = \\dfrac{2}{3} \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nf(t) = 2t^3 -2t^2 +q +\\dfrac{2}{3} \\ .\r\n\\]\r\n[3] \u3088\u308a, \\(\\alpha , \\beta , \\gamma\\) \u306e\u3044\u305a\u308c\u304b\u304c \\(\\pm \\dfrac{1}{\\sqrt{3}}\\) \u3067\u3042\u308a, \u5bfe\u79f0\u6027\u304b\u3089 \\(\\alpha = \\pm \\dfrac{1}{\\sqrt{3}}\\) \u3068\u3057\u3066\u3088\u3044.
    \r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf( \\pm \\alpha ) & = \\pm \\dfrac{2 \\sqrt{3}}{9} -\\dfrac{2}{3} +q +\\dfrac{2}{3} = 0 \\\\\r\n\\text{\u2234} \\quad q & = \\pm \\dfrac{2 \\sqrt{3} \\ . }{9}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, [1] \u3088\u308a\r\n\\[\r\np^3 -p = \\dfrac{8}{27} -\\dfrac{2}{3} = -\\dfrac{10}{27} \\gt q \\ .\r\n\\]\r\n\\(10^2 \\lt 6^2 \\cdot 3\\) \u3088\u308a \\(\\dfrac{10}{27} \\lt \\dfrac{2 \\sqrt{3}}{9}\\) \u306a\u306e\u3067, \\(q = -\\dfrac{2 \\sqrt{3}}{9}\\) \u306e\u307f\u304c\u9069\u3059\u308b.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b P \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( \\dfrac{2}{3} , -\\dfrac{2 \\sqrt{3}}{9} \\right)} \\ . \r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \\(3\\) \u6b21\u95a2\u6570 \\(y = x^3 -x\\) \u306e\u30b0\u30e9\u30d5\u3092 \\(C\\) \u3068\u3057, \u4e0d\u7b49\u5f0f \\[ x^3 -x \\gt y \\gt -x \\] \u306e\u8868\u3059\u9818\u57df\u3092 \\(D\\) \u3068\u3059\u308b. \u307e\u305f, P … \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":[131],"tags":[148,137],"class_list":["post-1285","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2015","tag-tohoku_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1285","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=1285"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1285\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1285"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1285"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1285"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}