{"id":1127,"date":"2015-06-22T10:05:23","date_gmt":"2015-06-22T01:05:23","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1127"},"modified":"2021-09-24T17:31:25","modified_gmt":"2021-09-24T08:31:25","slug":"tok201402","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201402\/","title":{"rendered":"\u6771\u5de5\u59272014\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(a \\gt 1\\) \u3068\u3057, \u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u8003\u3048\u308b.\r\n\\[\r\n\\text{(\uff0a)} \\quad \\dfrac{e^t -1}{t} \\geqq e^{\\frac{t}{a}}\r\n\\]\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a = 2\\) \u306e\u3068\u304d, \u3059\u3079\u3066\u306e \\(t \\gt 0\\) \u306b\u5bfe\u3057\u3066\u4e0a\u306e\u4e0d\u7b49\u5f0f (\uff0a) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3059\u3079\u3066\u306e \\(t \\gt 0\\) \u306b\u5bfe\u3057\u3066\u4e0a\u306e\u4e0d\u7b49\u5f0f (\uff0a) \u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    \\(t \\gt 0\\) \u306b\u304a\u3044\u3066, (\uff0a) \u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\ne^t -1 & \\geqq t e^{\\frac{t}{a}} \\\\\r\n\\text{\u2234} \\quad e^t -t e^{\\frac{t}{a}} -1 & \\geqq 0 \\quad ... [ \\text{A} ]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, (\uff0a) \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u3059\u306b\u306f, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
    \r\n[A] \u306e\u5de6\u8fba\u3092 \\(f(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf(0) = 1 -0 -1 = 0\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nf(t) \\ \\text{\u304c} \\ t \\gt 0 \\ \\text{\u306b\u304a\u3044\u3066, \u5358\u8abf\u5897\u52a0\u3067\u3042\u308b} \\quad ... [ \\text{B} ]\r\n\\]\r\n\u3067\u3042\u308c\u3070, [A] \u304c\u6210\u7acb\u3059\u308b.\r\n\\[\\begin{align}\r\nf'(t) & = e^t -e^{\\frac{t}{a}} -\\dfrac{t}{a} e^{\\frac{t}{a}} \\\\\r\n& = e^{\\frac{t}{a}} \\underline{\\left( e^{\\frac{(a-1) t}{a}} -\\dfrac{t}{a} -1 \\right)} _ {[1]}\r\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [1] \u3092 \\(g(t)\\) \u3068\u304a\u304f\u3068, \\(e^{\\frac{t}{a}} \\gt 0\\) \u306a\u306e\u3067, \\(f'(t)\\) \u3068 \\(g(t)\\) \u306e\u6b63\u8ca0\u306f\u4e00\u81f4\u3059\u308b.\r\n\\[\r\ng(0) = 1 -0 -1 = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\ng(t) \\ \\text{\u304c} \\ t \\gt 0 \\ \\text{\u306b\u304a\u3044\u3066, \u5358\u8abf\u5897\u52a0\u3067\u3042\u308b} \\quad ... [ \\text{C} ]\r\n\\]\r\n\u3067\u3042\u308c\u3070, [B] \u304c\u6210\u7acb\u3059\u308b.\r\n\\[\\begin{align}\r\ng'(t) & = \\dfrac{a-1}{a} e^{\\frac{(a-1) t}{a}} -\\dfrac{1}{a} \\\\\r\n& = \\dfrac{1}{a} \\left\\{ (a-1) e^{\\frac{(a-1) t}{a}} -1 \\right\\}\r\n\\end{align}\\]\r\n

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

    \\(a = 2\\) \u306e\u3068\u304d\r\n\\[\r\ng'(t) = \\dfrac{1}{2} \\left( e^{\\frac{t}{2}} -1 \\right) \\geqq 0\r\n\\]\r\n\u3088\u3063\u3066, [C] \u304c\u6210\u7acb\u3059\u308b\u306e\u3067, (\uff0a) \u3082\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\r\n

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

    \\(a\\) \u306e\u5024\u3067\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(a \\geqq 2\\) \u306e\u3068\u304d
      \r\n\\(a-1 \\geqq 1\\) \u306a\u306e\u3067\r\n\\[\r\ng'(t) = \\dfrac{1}{a} \\left\\{ (a-1) e^{\\frac{(a-1) t}{a}} -1 \\right\\} \\geqq 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [C] \u304c\u6210\u7acb\u3057, (\uff0a) \u3082\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(1 \\lt a \\lt 2\\) \u306e\u3068\u304d
      \r\n\\(g'(t) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\ne^{\\frac{(a-1) t}{a}} & = \\dfrac{1}{a-1} \\\\\r\n\\dfrac{a-1}{a} t & = \\log \\dfrac{1}{a-1} \\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{a}{a-1} \\log \\dfrac{1}{a-1}\r\n\\end{align}\\]\r\n\\(\\alpha = \\dfrac{a}{a-1} \\log \\dfrac{1}{a-1}\\) \u3068\u304a\u3051\u3070, \\(g(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} t & (0) & \\cdots & \\alpha & \\cdots & ( \\infty ) \\\\ \\hline g'(t) & & - & 0 & + & \\\\ \\hline g(t) & ( 0 ) & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & ( \\infty ) \\\\ \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g( \\beta ) = 0 \\quad ( \\alpha \\lt \\beta )\\) \u3092\u307f\u305f\u3059\u5b9f\u6570 \\(\\beta\\) \u304c\u5b58\u5728\u3059\u308b.
      \r\n\u3053\u306e\u3068\u304d\r\n\\[\r\nf'( \\beta ) = 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|cccc} t & (0) & \\cdots & \\beta & \\cdots \\\\ \\hline f'(t) & & - & 0 & + \\\\ \\hline f(t) & ( 0 ) & \\searrow & \\text{\u6700\u5c0f} & \\nearrow \\\\ \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \\(t \\gt 0\\) \u306b\u304a\u3044\u3066, [A] \u306f\u6210\u7acb\u305b\u305a, (\uff0a) \u3082\u6210\u7acb\u3057\u306a\u3044.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a, \u6c42\u3081\u308b \\(a\\) \u306e\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{a \\geqq 2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a \\gt 1\\) \u3068\u3057, \u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u8003\u3048\u308b. \\[ \\text{(\uff0a)} \\quad \\dfrac{e^t -1}{t} \\geqq e^{\\frac{t}{a}} \\] (1)\u3000\\(a = 2\\) \u306e\u3068\u304d, \u3059 … \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":[116],"tags":[141,112],"class_list":["post-1127","post","type-post","status-publish","format-standard","hentry","category-toko_2014","tag-toko","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1127","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=1127"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1127\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}