{"id":1113,"date":"2015-06-20T23:33:47","date_gmt":"2015-06-20T14:33:47","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1113"},"modified":"2025-03-18T01:48:13","modified_gmt":"2025-03-17T16:48:13","slug":"kyr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201404\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\r\n

\u5b9f\u6570\u306e\u5b9a\u6570 $a , b$ \u306b\u5bfe\u3057\u3066, \u95a2\u6570 $f(x)$ \u3092\r\n$$\r\nf(x) = \\dfrac{ax +b}{x^2 +x +1}\r\n$$\r\n\u3067\u5b9a\u3081\u308b. \u3059\u3079\u3066\u306e\u5b9f\u6570 $x$ \u3067\u4e0d\u7b49\u5f0f\r\n$$\r\nf(x) \\leqq {f(x)}^3 -2 {f(x)}^2 +2\r\n$$\r\n\u304c\u6210\u7acb\u3059\u308b\u3088\u3046\u306a\u70b9 $( a , b )$ \u306e\u7bc4\u56f2\u3092\u56f3\u793a\u305b\u3088. <\/p>\r\n

\r\n\r\n

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

$X = f(x)$ \u3068\u304a\u3051\u3070, \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3088\u308a\r\n$$\\begin{gather}\r\nX \\leqq X^3 -2X^2 +2 \\\\\r\n\\quad (X+1)(X-1)(X-2) \\geqq 0 \\\\\r\n\\therefore \\quad -1 \\leqq X \\leqq 1 , \\ 2 \\leqq X\r\n\\end{gather}$$\r\n\u3057\u305f\u304c\u3063\u3066, \u3059\u3079\u3066\u306e\u5b9f\u6570 $x$ \u306b\u3064\u3044\u3066\r\n$$\r\n-1 \\leqq f(x) \\leqq 1 , \\ 2 \\leqq f(x) \\quad ... \\maru{\\text{A}}\r\n$$\r\n\u3068\u306a\u308b\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n\u3068\u3053\u308d\u3067, $f(x)$ \u306e\u5206\u6bcd\u306b\u3064\u3044\u3066\r\n$$\r\nx^2 +x +1 = \\left( x +\\dfrac{1}{2} \\right)^2 +\\dfrac{3}{4} \\gt 0 \\quad ... \\maru{1}\r\n$$\r\n\u306a\u306e\u3067, $f(x)$ \u306f\u5b9f\u6570\u5168\u4f53\u3067\u9023\u7d9a\u3067\u3042\u308a,\r\n$$\r\n\\displaystyle\\lim _ {x \\rightarrow \\pm \\infty} f(x) = 0\r\n$$\r\n\u3067\u3042\u308b\u304b\u3089, $\\maru{\\text{A}}$ \u306e\u3046\u3061, $f(x) \\geqq 2$ \u3068\u306a\u308b\u3053\u3068\u306f\u306a\u3044.
\r\n\u3086\u3048\u306b, \u3059\u3079\u3066\u306e\u5b9f\u6570 $x$ \u306b\u3064\u3044\u3066\r\n$$\r\n-1 \\leqq f(x) \\leqq 1 \\quad ... \\maru{\\text{B}}\r\n$$\r\n\u3068\u306a\u308b\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n$\\maru{\\text{B}}$ \u3088\u308a\r\n$$\\begin{align}\r\n& -1 \\leqq \\dfrac{ax+b}{x^2+x+1} \\leqq 1 \\\\\r\n-x^2-x & -1 \\leqq ax+b \\leqq x^2+x+1 \\quad ( \\ \\because \\maru{1} \\ ) \\\\\r\n\\therefore & \\quad \\left\\{ \\begin{array}{ll} x^2 +(a+1)x +b+1 \\geqq 0 \\quad ... \\maru{2} \\\\ x^2 -(a-1)x -b+1 \\geqq 0 \\quad ... \\maru{3} \\end{array} \\right.\r\n\\end{align}$$\r\n$\\maru{2}$ \u304c\u5e38\u306b\u6210\u7acb\u3059\u308b\u6761\u4ef6\u306f, \u5224\u5225\u5f0f $D _ 1$ \u306b\u3064\u3044\u3066\r\n$$\\begin{align}\r\nD _ 1 & = (a+1)^2 -4(b+1) \\leqq 0 \\\\\r\n\\therefore \\quad b & \\geqq \\dfrac{(a+1)^2}{4} -1 \\quad ... \\maru{4}\r\n\\end{align}$$\r\n$\\maru{3}$ \u304c\u5e38\u306b\u6210\u7acb\u3059\u308b\u6761\u4ef6\u306f, \u5224\u5225\u5f0f $D _ 2$ \u306b\u3064\u3044\u3066\r\n$$\\begin{align}\r\nD _ 2 & = (a-1)^2 +4(b-1) \\leqq 0 \\\\\r\n\\therefore \\quad b & \\leqq -\\dfrac{(a-1)^2}{4} +1 \\quad ... \\maru{5}\r\n\\end{align}$$\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b $(a,b)$ \u306e\u6761\u4ef6\u306f $\\maru{4}$ \u304b\u3064 $\\maru{5}$ \u3067\u3042\u308a, \u56f3\u793a\u3059\u308b\u3068\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u542b\u3080\uff09\u3068\u306a\u308b. <\/p>\r\n

\"\" <\/p>","protected":false},"excerpt":{"rendered":"\u5b9f\u6570\u306e\u5b9a\u6570 $a , b$ \u306b\u5bfe\u3057\u3066, \u95a2\u6570 $f(x)$ \u3092 $$ f(x) = \\dfrac{ax +b}{x^2 +x +1} $$ \u3067\u5b9a\u3081\u308b. \u3059\u3079\u3066\u306e\u5b9f\u6570 $x$ \u3067\u4e0d\u7b49\u5f0f $$ f(x) \\leqq {f(x … \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":[114],"tags":[140,112],"class_list":["post-1113","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2014","tag-kyoto_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1113","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=1113"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1113\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1113"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1113"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}