{"id":939,"date":"2013-09-20T22:45:03","date_gmt":"2013-09-20T13:45:03","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=939"},"modified":"2021-10-23T03:20:37","modified_gmt":"2021-10-22T18:20:37","slug":"wsr201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201303\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(f(x) = \\dfrac{1}{2} e^{2x} +2e^{x} +x\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066 \\(g(x) = tx -f(x)\\) \u3068\u304a\u304f. \\(x\\) \u304c\u5b9f\u6570\u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, \\(g(x)\\) \u304c\u6700\u5927\u5024\u3092\u3082\u3064\u3088\u3046\u306a \\(t\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088. \u307e\u305f \\(t\\) \u304c\u305d\u306e\u7bc4\u56f2\u306b\u3042\u308b\u3068\u304d, \\(g(x)\\) \u306e\u6700\u5927\u5024\u3068\u305d\u306e\u3068\u304d\u306e \\(x\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u3067\u6c42\u3081\u305f\u6700\u5927\u5024\u3092 \\(m(t)\\) \u3068\u3059\u308b. \\(a\\) \u3092\u5b9a\u6570\u3068\u3057, \\(t\\) \u306e\u95a2\u6570 \\(h(t) = at -m(t)\\) \u3092\u8003\u3048\u308b. \\(t\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(h(t)\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \\(g(x)\\) \u3092\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\ng'(x) & = t -e^{2x} -2e^x -1 \\\\\r\n& = t -\\left( e^x+1 \\right)^2\r\n\\end{align}\\]\r\n\\(\\left( e^x+1 \\right)^2 \\geqq 0\\) \u306b\u6ce8\u610f\u3057\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(t \\leqq 0\\) \u306e\u3068\u304d\r\n\\[\r\ng'(x) \\leqq 0\r\n\\]\r\n\u306a\u306e\u3067, \\(g(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3068\u306a\u308a, \u6700\u5927\u5024\u3092\u3082\u305f\u306a\u3044.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(t \\gt 0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\ng'(x) & = \\left( \\sqrt{t} \\right)^2 -\\left( e^x+1 \\right)^2 \\\\\r\n& = -\\left( e^x +\\sqrt{t} +1 \\right) \\left( e^x -\\sqrt{t} +1 \\right)\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(e^x +\\sqrt{t} +1 \\gt 0\\) \u306a\u306e\u3067, \\(e^x +\\sqrt{t} +1\\) \u306b\u7740\u76ee\u3057\u3066, \u3055\u3089\u306b\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.\r\n

        \r\n

      1. (\u30a2)<\/strong>\u3000\\(0 \\lt t \\leqq 1\\) \u306e\u3068\u304d
        \r\n\\(e^x +\\sqrt{t} +1 \\gt 0\\) \u3088\u308a\r\n\\[\r\ng'(x) \\leqq 0\r\n\\]\r\n\u306a\u306e\u3067, \\(g(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3068\u306a\u308a, \u6700\u5927\u5024\u3092\u3082\u305f\u306a\u3044.<\/p><\/li>\r\n

      2. (\u30a4)<\/strong>\u3000\\(t \\gt 1\\) \u306e\u3068\u304d
        \r\n\\(g'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\ne^x & = \\sqrt{t} -1 \\\\\r\n\\text{\u2234} \\quad x & = \\log \\left( \\sqrt{t} -1 \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccc} x & \\cdots & \\log \\left( \\sqrt{t} -1 \\right) & \\cdots \\\\ \\hline g'(x) & + & 0 & - \\\\ \\hline g(x) & \\nearrow & \\text{\u6700\u5927} & \\searrow \\end{array}\r\n\\]\r\n\u3053\u306e\u3068\u304d, \u6700\u5927\u5024\u306f\r\n\\[\\begin{align}\r\ng & \\left( \\log \\left( \\sqrt{t} -1 \\right) \\right) \\\\\r\n& = t \\log \\left( \\sqrt{t} -1 \\right) -\\dfrac{\\left( \\sqrt{t} -1 \\right)^2}{2} -2 \\left( \\sqrt{t} -1 \\right) -\\log \\left( \\sqrt{t} -1 \\right) \\\\\r\n& = (t-1) \\log \\left( \\sqrt{t} -1 \\right) -\\dfrac{t}{2} -\\sqrt{t} +\\dfrac{3}{2}\r\n\\end{align}\\]<\/li>\r\n<\/ol><\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(t\\) \u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{t \\gt 1}\r\n\\]\r\n\u3067\u3042\u308a, \u3053\u306e\u3068\u304d\u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{(t-1) \\log \\left( \\sqrt{t} -1 \\right) -\\dfrac{t}{2} -\\sqrt{t} +\\dfrac{3}{2}}\r\n\\]\r\n

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

        \\(h(t)\\) \u3092\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nh'(t) & = a -\\left\\{ \\log \\left( \\sqrt{t} -1 \\right) +\\left( \\sqrt{t} -1 \\right) \\cdot \\dfrac{\\frac{1}{2 \\sqrt{t}}}{\\sqrt{t} -1} -\\dfrac{1}{2} -\\dfrac{1}{2 \\sqrt{t}} \\right\\} \\\\\r\n& = a -\\left\\{ \\log \\left( \\sqrt{t} -1 \\right) +\\dfrac{\\left( \\sqrt{t} +1 \\right) -\\sqrt{t} -1}{2 \\sqrt{t}} \\right\\} \\\\\r\n& = a -\\log \\left( \\sqrt{t} -1 \\right)\r\n\\end{align}\\]\r\n\\(h'(t) = 0\\) \u3092\u3068\u304f\u3068, \\(t \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ne^a & = \\sqrt{t} -1 \\\\\r\n\\text{\u2234} \\quad t & = \\left( e^a +1 \\right)^2\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \\(h(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} t & (1) & \\cdots & \\left( e^a +1 \\right)^2 & \\cdots \\\\ \\hline h'(t) & & + & 0 & - \\\\ \\hline h(t) & & \\nearrow & \\text{\u6700\u5927} & \\searrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\\begin{align}\r\nh & \\left( \\left( e^a +1 \\right)^2 \\right) \\\\\r\n& = a \\left( e^a +1 \\right)^2 -\\left[ \\left\\{ \\left( e^a +1 \\right)^2 -1 \\right\\} a -\\dfrac{\\left( e^a +1 \\right)^2}{2} -\\left( e^a +1 \\right) +\\dfrac{3}{2} \\right] \\\\\r\n& = a +\\dfrac{1 +2 e^a +e^{2a}}{2} +e^a +1 -\\dfrac{3}{2} \\\\\r\n& = \\underline{\\dfrac{1}{2} e^{2a} +2e^a +a}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = \\dfrac{1}{2} e^{2x} +2e^{x} +x\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066 \\(g(x) = tx -f(x)\\) \u3068\u304a\u304f. \\(x\\) \u304c\u5b9f\u6570\u5168\u4f53 … \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":[100],"tags":[147,111],"class_list":["post-939","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2013","tag-waseda_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/939","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=939"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/939\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=939"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=939"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}