{"id":456,"date":"2012-11-01T23:17:24","date_gmt":"2012-11-01T14:17:24","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=456"},"modified":"2021-10-03T15:54:57","modified_gmt":"2021-10-03T06:54:57","slug":"kbr201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201001\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(f(x) = \\dfrac{1}{3} x^3 -\\dfrac{1}{2} ax^2\\) \u3068\u304a\u304f.\r\n\u305f\u3060\u3057 \\(a \\gt 0\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f(-1) \\leqq f(3)\\) \u3068\u306a\u308b \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(f(x)\\) \u306e\u6975\u5c0f\u5024\u306f \\(f(-1)\\) \u4ee5\u4e0b\u3068\u306a\u308b \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(-1 \\leqq x \\leqq 3\\) \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u6700\u5c0f\u5024\u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\[\\begin{align}\r\nf(-1) & = -\\dfrac{1}{3} -\\dfrac{a}{2} \\\\\r\nf(3) & = 9 -\\dfrac{9a}{2}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(f(-1) \\leqq f(3)\\) \u3088\u308a\r\n\\[\\begin{align}\r\n-\\dfrac{1}{3} -\\dfrac{a}{2} & \\leqq 9 -\\dfrac{9a}{2} \\\\\r\n4a & \\leqq \\dfrac{28}{3} \\\\\r\n\\text{\u2234} \\quad a & \\leqq \\dfrac{7}{3}\r\n\\end{align}\\]\r\n\\(a \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\underline{0 \\lt a \\leqq \\dfrac{7}{3}}\r\n\\]\r\n

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

    \\[\r\nf'(x) = x^2-ax = x(x-a)\r\n\\]\r\n\u306a\u306e\u3067, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & \\cdots & 0 & \\cdots & a & \\cdots \\\\ \\hline f'(x) & + & 0 & - & 0 & + \\\\ \\hline f(x) & \\nearrow & 0 & \\searrow & -\\dfrac{a^3}{6} & \\nearrow \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(a) \\leqq f(-1)\\) \u3088\u308a\r\n\\[\\begin{align}\r\n-\\dfrac{a^3}{6} \\leqq -\\dfrac{1}{3} & -\\dfrac{a}{2} \\\\\r\na^3 -3a -2 & \\geqq 0 \\\\\r\n(a-2)(a+1)^2 & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad \\underline{a \\geqq 2} &\r\n\\end{align}\\]\r\n

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

    \\(-1 \\leqq x \\leqq 3\\) \u306b\u304a\u3044\u3066\u6700\u5c0f\u5024\u3068\u306a\u308a\u3046\u308b\u5019\u88dc\u306f\r\n\\[\r\nf(-1) , \\ f(3) , \\ f(a) \\ ( \\ -1\\leq a \\leqq 3 \\text{\u306e\u3068\u304d\u306b\u9650\u308b} )\r\n\\]\r\n\u306e \\(3\\) \u3064\u3067\u3042\u308b.
    \r\n(1) (2)<\/strong> \u3067, \\(f(-1)\\) \u3068 \\(f(3)\\) , \\(f(-1)\\) \u3068 \\(f(a)\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3057\u305f\u306e\u3067, \u3053\u3053\u3067, \\(f(3)\\)\u3068 \\(f(a)\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b.
    \r\n\\(f(a) \\leqq f(3)\\) \u3088\u308a\r\n\\[\\begin{align}\r\n-\\dfrac{a^3}{6} & \\leqq 9 -\\dfrac{9a}{2} \\\\\r\na^3 -27a +54 & \\geqq 0 \\\\\r\n(a+6)(a-3)^2 & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad a & = 3 \\quad ( \\ \\text{\u2235} \\ a \\gt 0 \\ )\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\left\\{ \\begin{array}{ll} -\\dfrac{1}{3} -\\dfrac{a}{2} & \\left( \\ 0 \\lt a \\lt 2 \\text{\u306e\u3068\u304d} \\right) \\\\ -\\dfrac{a^3}{6} & \\left( \\ 2 \\leqq a \\lt 3 \\text{\u306e\u3068\u304d} \\right) \\\\ 9 -\\dfrac{9a}{2} & \\left( \\ a \\geqq 3 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = \\dfrac{1}{3} x^3 -\\dfrac{1}{2} ax^2\\) \u3068\u304a\u304f. \u305f\u3060\u3057 \\(a \\gt 0\\) \u3068\u3059\u308b. (1)\u3000\\(f(-1) \\leqq f(3)\\) \u3068\u306a\u308b \\(a\\) \u306e\u7bc4 … \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":[40],"tags":[144,14],"class_list":["post-456","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2010","tag-tsukuba_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/456","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=456"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/456\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}