{"id":1263,"date":"2015-09-03T22:57:28","date_gmt":"2015-09-03T13:57:28","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1263"},"modified":"2021-09-13T19:13:56","modified_gmt":"2021-09-13T10:13:56","slug":"iks201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201502\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \\(f(x) = x^3 -3ax +b\\) \u3068\u304a\u304f.\r\n\\(-1 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b \\(\\left| f(x) \\right|\\) \u306e\u6700\u5927\u5024\u3092 \\(M\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a \\gt 0\\) \u306e\u3068\u304d, \\(f(x)\\) \u306e\u6975\u5024\u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(b \\geqq 0\\) \u306e\u3068\u304d, \\(M\\) \u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a , b\\) \u304c\u5b9f\u6570\u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, \\(M\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\r\nf'(x) = 3 ( x^2 -a ) \\ .\r\n\\]\r\n\\(a \\gt 0\\) \u306a\u306e\u3067, \\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\pm \\sqrt{a} \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} x & \\cdots & -\\sqrt{a} & \\cdots & \\sqrt{a} & \\cdots \\\\ \\hline f'(x) & + & 0 & - & 0 & + \\\\ \\hline f(x) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6975\u5024\u306f\r\n\\[\r\nf \\left( \\pm \\sqrt{a} \\right) = \\underline{\\mp 2a \\sqrt{a} +b \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\quad } \\ .\r\n\\]\r\n

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

    \\[\r\nf( \\pm 1 ) = \\pm ( 1-3a ) +b \\ .\r\n\\]\r\n\\(a \\leqq 0\\) \u306e\u3068\u304d, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3059\u308b\u3053\u3068, \u307e\u305f \\(b \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n-f( \\pm1 ) = \\mp (1-3a) -b \\leqq f( \\mp 1 ) \\ .\r\n\\]\r\n\u306b\u6ce8\u610f\u3059\u308c\u3070, \\(M\\) \u306e\u5019\u88dc\u306f\r\n\\[\\begin{gather}\r\nf(1) = -3a+b+1 , \\quad f(-1) = 3a+b-1 , \\\\\r\nf \\left( -\\sqrt{a} \\right) = 2a \\sqrt{a} +b \\quad ( \\ 0 \\lt a \\leqq 1 \\ \\text{\u306e\u3068\u304d\u306e\u307f} ) \\ .\r\n\\end{gather}\\]\r\n\u3053\u308c\u3089\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068, \u4e0b\u306e\u30b0\u30e9\u30d5\u306e\u3088\u3046\u306b\u306a\u308a<\/p>\r\n\"iks20150201\"\r\n

    \u3088\u3063\u3066\r\n\\[\r\nM = \\underline{\\left\\{ \\begin{array}{ll} -3a+b+1 & \\left( \\ a \\lt \\dfrac{1}{4} \\ \\text{\u306e\u3068\u304d} \\right) \\\\ 2a \\sqrt{a} +b & \\left( \\ \\dfrac{1}{4} \\leqq a \\lt 1 \\ \\text{\u306e\u3068\u304d} \\right) \\\\ 3a+b-1 & ( \\ a \\geqq 1 \\ \\text{\u306e\u3068\u304d} ) \\end{array} \\right.} \\ .\r\n\\]\r\n

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

    \\(b \\lt 0\\) \u306e\u3068\u304d, \\(b' = -b \\ ( \\gt 0 )\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\left| f(-x) \\right| = \\left| -( x^3 -3ax +b' ) \\right| = \\left| f(x) \\right| \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(b = k \\ ( \\geqq 0 )\\) \u306e\u3068\u304d\u306e \\(y = \\left| f(x) \\right|\\) \u306e\u30b0\u30e9\u30d5\u3068, \\(b = -k\\) \u306e\u3068\u304d\u306e \\(y = \\left| f(x) \\right|\\) \u306e\u30b0\u30e9\u30d5\u306f, \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, \\(M\\) \u306e\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f, \\(b \\geqq 0\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u304f, (2)<\/strong> \u306e\u7d50\u679c\u304b\u3089\r\n\\[\r\nM \\geqq b +\\dfrac{1}{4} \\geqq \\dfrac{1}{4} \\ .\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\underline{M \\geqq \\dfrac{1}{4}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \\(f(x) = x^3 -3ax +b\\) \u3068\u304a\u304f. \\(-1 \\leqq x \\leqq 1\\) \u306b\u304a\u3051\u308b \\(\\left| f(x) \\right|\\) \u306e\u6700\u5927\u5024\u3092 \\(M\\) … \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":[130],"tags":[145,137],"class_list":["post-1263","post","type-post","status-publish","format-standard","hentry","category-ikashika_2015","tag-ikashika","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1263","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=1263"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1263\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}