\r\n \r\n
\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
(1)<\/strong><\/p>\r\n\\(F_c = \\left( 1 +\\dfrac{1}{x} \\right) \\left( 1 +\\dfrac{1}{y} \\right)\\) \u3068\u304a\u304f.
\r\n\u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nF_c & = 1 +\\dfrac{1}{x} +\\dfrac{1}{y} +\\dfrac{1}{xy} \\\\\r\n& = 1 +\\dfrac{c+1}{xy} \\quad ( \\ \\text{\u2235} \\ x+y = c \\ ) \\\\\r\n& \\geqq 1 +(c+1) \\left( \\dfrac{2}{x+y} \\right)^2 \\\\\r\n& = 1 +\\dfrac{4 (c+1)}{c^2} \\\\\r\n& = \\left( \\dfrac{c+2}{c} \\right)^2\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f \\(x = y = \\dfrac{c}{2}\\) \u306e\u3068\u304d.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\left( \\dfrac{c+2}{c} \\right)^2}\r\n\\]\r\n
(2)<\/strong><\/p>\r\n\\(G = \\left( 1 +\\dfrac{1}{x} \\right) \\left( 1 +\\dfrac{1}{y} \\right) \\left( 1 -\\dfrac{4}{3z} \\right)\\) \u3068\u304a\u304f.
\r\n\u307e\u305a \\(z\\) \u3092\u5b9a\u6570\u3068\u307f\u3066\u8003\u3048\u308b.
\r\n\u6761\u4ef6\u3088\u308a \\(x+y = 1-z \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\nG = F _{1-z} \\underline{\\left( 1 -\\dfrac{4}{3z} \\right)} _{[1]}\r\n\\]\r\n\\(0 \\lt z \\lt 1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{4}{3z} & \\gt \\dfrac{4}{3} \\\\\r\n\\text{\u2234} \\quad [1] & \\lt 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(G\\) \u306f\u8ca0\u3067, \\(F _{1-z}\\) \u304c\u6700\u5c0f\u306e\u3068\u304d, \\(G\\) \u306f\u6700\u5927\u5024 \\(f(z)\\) \u3092\u3068\u308b.
\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nF _{1-z} \\geqq \\left( \\dfrac{1+z}{1-z} \\right)^2 = \\left( 1 +\\dfrac{2}{1-z} \\right)^2\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nf(z) = \\left( 1 -\\dfrac{4}{3z} \\right) \\left( 1 +\\dfrac{2}{1-z} \\right)^2\r\n\\]\r\n\u3053\u3053\u3067, \\(z\\) \u3092\u5909\u6570\u3068\u307f\u3066\r\n\\[\\begin{align}\r\nf'(z) & = \\dfrac{4}{3 z^2} \\left( 1 +\\dfrac{2}{1-z} \\right)^2 \\\\\r\n& \\qquad +\\left( 1 -\\dfrac{4}{3z} \\right) \\cdot 2 \\left( 1 +\\dfrac{2}{1-z} \\right) \\dfrac{2}{(1-z)^2} \\\\\r\n& = \\dfrac{4 (3-z)^2 +4z (3-z) (3z-4)}{3 z^2 (1-z)^3} \\\\\r\n& = \\dfrac{4 (3-z) ( 4z^2 -8z +3 )}{3 z^2 (1-z)^3} \\\\\r\n& = \\dfrac{4 (3-z) (2z-3) (2z-1)}{3 z^2 (1-z)^3}\r\n\\end{align}\\]\r\n\\(0 \\lt z \\lt 1\\) \u306b\u304a\u3044\u3066 \\(f'(z) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nz = \\dfrac{1}{2}\r\n\\]\r\n\u3053\u306e\u7bc4\u56f2\u306b\u304a\u3044\u3066 \\(3-z \\gt 0\\) , \\(2z-3 \\lt 0\\) \u306b\u6ce8\u610f\u3059\u308c\u3070, \\(f(z)\\) \u306e\u5897\u6e1b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} z & (0) & \\cdots & \\dfrac{1}{2} & \\cdots & (1) \\\\ \\hline f'(z) & & + & 0 & - & \\\\ \\hline f(z) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(G\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\nf \\left( \\dfrac{1}{2} \\right) = -\\dfrac{5}{3} \\cdot 5^2 = \\underline{-\\dfrac{125}{3}}\r\n\\]\r\n\r\n \r\n « \u89e3\u7b54\u3092\u96a0\u3059 <\/a>\r\n <\/p>\r\n <\/div>","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(c\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \u6b63\u306e\u5b9f\u6570 \\(x , y\\) \u304c \\(x+y = c\\) \u3092\u307f\u305f\u3059\u3068\u304d, \\[ \\left( 1 +\\dfrac{1}{x} \\right) \\left( 1 […]","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":[152],"tags":[142,162],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1438"}],"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=1438"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1438\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1438"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1438"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}