{"id":169,"date":"2011-12-02T22:16:43","date_gmt":"2011-12-02T13:16:43","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=169"},"modified":"2021-03-16T16:16:21","modified_gmt":"2021-03-16T07:16:21","slug":"tkr200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200905\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2009\uff1a\u7b2c5\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u5b9f\u6570 \\(x\\) \u304c \\(-1 \\lt x \\lt 1\\) , \\(x \\neq 0\\) \u3092\u307f\u305f\u3059\u3068\u304d, \u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u305b.\r\n\\[\r\n(1-x)^{1-\\frac{1}{x}} \\lt (1+x)^{\\frac{1}{x}}\r\n\\]<\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u305b.\r\n\\[\r\n0.9999^{101} \\lt 0.99 \\lt 0.9999^{100}\r\n\\]<\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\n& (1-x)^{1-\\frac{1}{x}} \\lt (1+x)^{\\frac{1}{x}} \\\\\r\n& \\Longleftrightarrow \\left( 1-\\dfrac{1}{x} \\right) \\log (1-x) \\lt \\dfrac{1}{x} \\log (1+x) \\\\\r\n& \\Longleftrightarrow \\left\\{ \\begin{array}{ll} (x-1) \\log (1-x) \\lt \\log (1+x) & \\left( 0 \\lt x \\lt 1 \\text{\u306e\u3068\u304d} \\right) \\\\ (x-1) \\log (1-x) \\gt \\log (1+x) & \\left( -1 \\lt x \\lt 0 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right. \\quad ... [\\text{\uff0a}]\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, [\uff0a] \u3092\u793a\u305b\u3070\u3088\u3044.<br \/>\r\n\\(f(x) = \\log (1+x) +(1-x) \\log (1-x)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{1+x} -\\log (1-x) -1 , \\\\\r\nf''(x) & = - \\dfrac{1}{(1+x)^2} +\\dfrac{1}{1-x} \\\\\r\n& = \\dfrac{-(1-x) +(1+x)^2}{(1-x)(1+x)^2} \\\\\r\n& = \\dfrac{x(x+3)}{(1-x)(1+x)^2}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\nf'(0) = 1 -\\log 1 -1 =0 , \\quad f(0) = \\log 1 + 1 \\cdot \\log 1 =0\r\n\\]\r\n\u306a\u306e\u3067, \\(-1 \\lt x \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} x & (-1) & \\cdots & 0 & \\cdots & (1) \\\\ \\hline f''(x) & & - & 0 & + & \\\\ \\hline f'(x) & & + & 0 & + \\\\ \\hline f(x) & & - & 0 & + & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [\uff0a] \u306f\u6210\u7acb\u3057\u3066\u3044\u308b.<br \/>\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u7d50\u679c\u306b, \\(t = 0.01\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n0.99^{-99} & \\lt 1.01^{100} \\\\\r\n0.99^{-99} \\cdot 0.99^{100} & \\lt \\left( 0.99 \\cdot 1.01 \\right)^{100} \\\\\r\n\\text{\u2234} \\quad 0.99 & \\lt 0.9999^{100} \\quad ... [1]\r\n\\end{align}\\]\r\n\u307e\u305f, \\(t = -0.01\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n1.01^{101} & \\lt 0.99^{-100} \\\\\r\n\\left( 1.01 \\cdot 0.99 \\right)^{101} & \\lt 0.99^{-100} \\cdot 0.99^{101} \\\\\r\n\\text{\u2234} \\quad 0.9999^{101} & \\lt 0.99 \\quad ... [2]\r\n\\end{align}\\]\r\n[1] [2] \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\u5b9f\u6570 \\(x\\) \u304c \\(-1 \\lt x \\lt 1\\) , \\(x \\neq 0\\) \u3092\u307f\u305f\u3059\u3068\u304d, \u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u305b. \\[ (1-x)^{1-\\frac{1}{x}} \\lt (1+x)^{\\frac{1 &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tkr200905\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[18],"tags":[139,15],"class_list":["post-169","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2009","tag-tokyo_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/169","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=169"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/169\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=169"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=169"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}