{"id":1173,"date":"2015-07-21T23:46:05","date_gmt":"2015-07-21T14:46:05","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1173"},"modified":"2021-09-29T23:00:50","modified_gmt":"2021-09-29T14:00:50","slug":"tbr201401","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201401\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2014\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(f(x) = x^3 -x\\) \u3068\u3059\u308b.\r\n\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306b\u70b9 \\(P \\ ( a , b )\\) \u304b\u3089\u5f15\u3044\u305f\u63a5\u7dda\u306f \\(3\\) \u672c\u3042\u308b\u3068\u3059\u308b.\r\n\\(3\\) \u3064\u306e\u63a5\u70b9 A \\(\\left( \\alpha , f( \\alpha ) \\right)\\) , B \\(\\left( \\beta , f( \\beta ) \\right)\\) , C \\(\\left( \\gamma , f( \\gamma ) \\right)\\) \u3092\u9802\u70b9\u3068\u3059\u308b\u4e09\u89d2\u5f62\u306e\u91cd\u5fc3\u3092 \\(G\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\alpha +\\beta +\\gamma\\) , \\(\\alpha \\beta +\\beta \\gamma +\\gamma \\alpha\\) \u304a\u3088\u3073 \\(\\alpha \\beta \\gamma\\) \u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 \\(G\\) \u306e\u5ea7\u6a19\u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u70b9 \\(G\\) \u306e \\(x\\) \u5ea7\u6a19\u304c\u6b63\u3067, \\(y\\) \u5ea7\u6a19\u304c\u8ca0\u3068\u306a\u308b\u3088\u3046\u306a\u70b9 \\(P\\) \u306e\u7bc4\u56f2\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n

    \u89e3\u7b54<\/h3>\r\n

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

    \\[\r\nf'(x) = 3x^2 -1\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 \\(\\left( t , f(t) \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\left( 3t^2 -1 \\right) (x-t) +t^3 -t \\\\\r\n& = \\left( 3t^2 -1 \\right) x -2 t^2\r\n\\end{align}\\]\r\n\u3053\u308c\u304c \\(P\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{gather}\r\nb = \\left( 3t^2 -1 \\right) a -2 t^2 \\\\\r\n\\text{\u2234} \\quad 2t^3 -3a t^2 +a +b = 0 \\quad ... [1]\r\n\\end{gather}\\]\r\n\\(\\alpha , \\beta , \\gamma\\) \u306f, \\(t\\) \u306b\u3064\u3044\u3066\u306e \\(3\\) \u6b21\u65b9\u7a0b\u5f0f [1] \u306e\u7570\u306a\u308b \\(3\\) \u3064\u306e\u89e3\u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{l} \\alpha +\\beta +\\gamma = \\dfrac{3a}{2} \\\\ \\alpha \\beta +\\beta \\gamma +\\gamma \\alpha = 0 \\\\ \\alpha \\beta \\gamma = -\\dfrac{a+b}{2} \\end{array} \\right.}\r\n\\]\r\n

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

    \\(G \\ \\left( \\dfrac{\\alpha +\\beta +\\gamma}{3} , \\dfrac{f( \\alpha ) +f( \\beta ) +f( \\gamma )}{3} \\right)\\) \u3068\u8868\u305b\u308b.
    \r\n\u3053\u3053\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3082\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nf( \\alpha ) & +f( \\beta ) +f( \\gamma ) \\\\\r\n& = \\left( {\\alpha}^3 +{\\beta}^3 +{\\gamma}^3 \\right) -( \\alpha +\\beta +\\gamma ) \\\\\r\n& = ( \\alpha +\\beta +\\gamma ) \\left\\{ ( \\alpha +\\beta +\\gamma )^2 -3 ( \\alpha \\beta +\\beta \\gamma +\\gamma \\alpha ) \\right\\} \\\\\r\n& \\qquad +3 \\alpha \\beta \\gamma -( \\alpha +\\beta +\\gamma ) \\\\\r\n& = \\dfrac{3a}{2} \\left\\{ \\left( \\dfrac{3a}{2} \\right)^2 -3 \\cdot 0 \\right\\} -\\dfrac{3 (a+b)}{2} -\\dfrac{3a}{2} \\\\\r\n& = \\dfrac{27 a^3}{8} -3a -\\dfrac{3b}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nG \\ \\underline{\\left( \\dfrac{a}{2} , \\dfrac{27 a^3}{8} -3a -\\dfrac{3b}{2} \\right)}\r\n\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{a}{2} \\gt 0 & , \\ \\dfrac{27 a^3}{8} -3a -\\dfrac{3b}{2} \\lt 0 \\\\\r\n\\text{\u2234} \\quad a \\gt 0 & , \\ b \\gt \\dfrac{9 a^3}{4} -2a \\quad ... [2]\r\n\\end{align}\\]\r\n\u3042\u3089\u305f\u3081\u3066, \\(P\\) \u304b\u3089 \\(3\\) \u672c\u306e\u63a5\u7dda\u3092\u5f15\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u6761\u4ef6\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
    \r\n[1] \u304c\u7570\u306a\u308b \\(3\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3066\u3070\u3088\u3044\u306e\u3067, [1] \u306e\u5de6\u8fba\u3092 \\(g(t)\\) \u3068\u304a\u3051\u3070\r\n\\[\r\ng'(t) = 6t^2 -6at = 6t (t-a)\r\n\\]\r\n\\(a \\neq 0\\) ... [3] \u3067\u3042\u308c\u3070 \\(g(t)\\) \u306f \\(t = 0 , a\\) \u3067\u6975\u5024\u3092\u3068\u308a, \u8003\u3048\u305f\u3044\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\ng(0) g(a) \\lt 0 \\\\\r\n\\text{\u2234} \\quad (a+b) \\left( -a^3 +a +b \\right) & \\lt 0 \\quad ... [4]\r\n\\end{align}\\]\r\n[2] \uff5e [4] \u3088\u308a, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n\"tbr20140101\"\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = x^3 -x\\) \u3068\u3059\u308b. \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306b\u70b9 \\(P \\ ( a , b )\\) \u304b\u3089\u5f15\u3044\u305f\u63a5\u7dda\u306f \\(3\\) \u672c\u3042\u308b\u3068\u3059\u308b. \\(3\\) \u3064\u306e\u63a5\u70b9 A \\(\\left( \\alp … \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":[121],"tags":[144,112],"class_list":["post-1173","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2014","tag-tsukuba_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1173","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=1173"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1173\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}