{"id":1195,"date":"2015-07-29T09:57:08","date_gmt":"2015-07-29T00:57:08","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1195"},"modified":"2021-10-23T02:57:23","modified_gmt":"2021-10-22T17:57:23","slug":"wsr201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201402\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52014\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(3\\) \u6b21\u95a2\u6570 \\(f(x) = x^3 -ax -b\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a \\gt 0\\) \u3067\u3042\u308b\u3068\u304d, \\(f(x)\\) \u306e\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6b21\u306e (i)<\/strong> , (ii)<\/strong> , (iii)<\/strong> \u3092\u793a\u305b.\r\n

      \r\n

    1. (i)<\/strong>\u3000\\(27b^2 -4a^3 \\gt 0\\) \u306e\u3068\u304d, \\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306f\u305f\u3060 \\(1\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064.<\/p><\/li>\r\n

    2. (ii)<\/strong>\u3000\\(27b^2 -4a^3 = 0\\) \u304b\u3064 \\(a \\gt 0\\) \u306e\u3068\u304d, \\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306f\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064.<\/p><\/li>\r\n

    3. (iii)<\/strong>\u3000\\(27b^2 -4a^3 \\lt 0\\) \u306e\u3068\u304d, \\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306f\u7570\u306a\u308b \\(3\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064.<\/p><\/li>\r\n<\/ol><\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

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

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

      \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nf'(x) & = 3 x^2 -a \\\\\r\n& = \\left( \\sqrt{3} x +\\sqrt{a} \\right) \\left( \\sqrt{3} x -\\sqrt{a} \\right)\r\n\\end{align}\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = \\pm \\sqrt{\\dfrac{a}{3}}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & \\cdots & -\\sqrt{\\dfrac{a}{3}} & \\cdots & \\sqrt{\\dfrac{a}{3}} & \\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\r\n\\[\\begin{align}\r\n\\text{\u6975\u5927\u5024} : \\ f \\left( -\\sqrt{\\dfrac{a}{3}} \\right) & = -\\dfrac{a}{3} \\sqrt{\\dfrac{a}{3}} +a \\sqrt{\\dfrac{a}{3}} -b \\\\\r\n& = \\underline{\\dfrac{2 \\sqrt{3}}{9} a^{\\frac{3}{2}} -b} \\\\\r\n\\text{\u6975\u5c0f\u5024} : \\ f \\left( \\sqrt{\\dfrac{a}{3}} \\right) & = \\dfrac{a}{3} \\sqrt{\\dfrac{a}{3}} -a \\sqrt{\\dfrac{a}{3}} -b \\\\\r\n& = \\underline{-\\dfrac{2 \\sqrt{3}}{9} a^{\\frac{3}{2}} -b}\r\n\\end{align}\\]\r\n

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

      \\(\\displaystyle\\lim _ {x \\rightarrow \\pm \\infty} f(x) = \\pm \\infty\\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b.<\/p>\r\n

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

        \r\n

      1. 1*<\/strong>\u3000\\(a \\leqq 0\\) \u306e\u3068\u304d\r\n\\[\r\nf'(x) = 3x^2 -a \\geqq 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057, \\(f(x)\\) \u3068 \\(x\\) \u8ef8\u306f\u305f\u3060 \\(1\\) \u70b9\u3092\u5171\u6709\u3059\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(a \\gt 0\\) \u306e\u3068\u304d
        \r\n\u6975\u5927\u5024\u306b\u3064\u3044\u3066\r\n\\[\r\nf \\left( -\\sqrt{\\dfrac{a}{3}} \\right) = \\dfrac{\\sqrt{3}}{9} \\left( \\sqrt{4 a^3} -\\sqrt{27 b^2} \\right) \\lt 0\r\n\\]\r\n\u3086\u3048\u306b, \\(f(x)\\) \u3068 \\(x\\) \u8ef8\u306f\u305f\u3060 \\(1\\) \u70b9\u3092\u5171\u6709\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

        \u6975\u5927\u5024\u306b\u3064\u3044\u3066\r\n\\[\r\nf \\left( -\\sqrt{\\dfrac{a}{3}} \\right) = \\dfrac{\\sqrt{3}}{9} \\left( \\sqrt{4 a^3} -\\sqrt{27 b^2} \\right) = 0\r\n\\]\r\n\u3086\u3048\u306b, \\(f(x)\\) \u3068 \\(x\\) \u8ef8\u306f\u7570\u306a\u308b \\(2\\) \u70b9\u3092\u5171\u6709\u3059\u308b.
        \r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

        \u6761\u4ef6\u3088\u308a, \\(0 \\leqq 27 b^2 \\lt 4 a^3\\) \u306a\u306e\u3067\r\n\\[\r\na \\gt 0\r\n\\]\r\n\u6975\u5024\u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nf \\left( -\\sqrt{\\dfrac{a}{3}} \\right) f \\left( \\sqrt{\\dfrac{a}{3}} \\right) & = b^2 -\\dfrac{4}{27} a^3 \\\\\r\n& = \\dfrac{27 b^2 -4 a^3}{27} \\lt 0\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u306e\u7b26\u53f7\u304c\u7570\u306a\u308b\u3053\u3068\u304b\u3089, \\(f(x)\\) \u3068 \\(x\\) \u8ef8\u306f\u7570\u306a\u308b \\(3\\) \u70b9\u3092\u5171\u6709\u3059\u308b.
        \r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(3\\) \u6b21\u95a2\u6570 \\(f(x) = x^3 -ax -b\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(a \\gt 0\\) \u3067\u3042\u308b\u3068\u304d, \\(f(x)\\) \u306e\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u3092\u6c42\u3081\u3088. (2)\u3000\u6b21\u306e (i) , (ii) … \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":[120],"tags":[147,112],"class_list":["post-1195","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2014","tag-waseda_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1195","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=1195"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1195\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}