{"id":29,"date":"2011-11-25T21:44:16","date_gmt":"2011-11-25T12:44:16","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=29"},"modified":"2021-09-09T20:43:19","modified_gmt":"2021-09-09T11:43:19","slug":"osr201103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201103\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2011\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570\u306e\u7d44 \\(( p , q )\\) \u306b\u5bfe\u3057, \\(f(x) = (x-p)^2 +q\\) \u3068\u304a\u304f.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u653e\u7269\u7dda \\(y = f(x)\\) \u304c\u70b9 \\(( 0 , 1 )\\) \u3092\u901a\u308a, \u3057\u304b\u3082\u76f4\u7dda \\(y = x\\) \u306e \\(x \\gt 0\\) \u306e\u90e8\u5206\u3068\u63a5\u3059\u308b\u3088\u3046\u306a\u5b9f\u6570\u306e\u7d44 \\(( p , q )\\) \u3068\u63a5\u70b9\u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u5b9f\u6570\u306e\u7d44 \\(( p _ 1 , q _ 1 ) , \\ ( p _ 2 , q _ 2 )\\) \u306b\u5bfe\u3057\u3066, \\(f _ 1(x) = ( x-p _ 1 )^2 +q _ 1\\) \u304a\u3088\u3073 \\(f _ 2(x) = ( x-p _ 2 )^2 +q _ 2\\) \u3068\u304a\u304f. \u5b9f\u6570 \\(\\alpha , \\beta\\) \uff08\u305f\u3060\u3057 \\(\\alpha \\lt \\beta\\) \uff09\u306b\u5bfe\u3057\u3066\r\n\\[\r\nf _ 1( \\alpha ) \\lt f _ 2( \\alpha ) \\ \\text{\u304b\u3064} \\ f _ 1( \\beta ) \\lt f _ 2( \\beta )\r\n\\]\r\n\u3067\u3042\u308b\u306a\u3089\u3070, \u533a\u9593 \\(\\alpha \\leqq x \\leqq \\beta\\) \u306b\u304a\u3044\u3066\u4e0d\u7b49\u5f0f \\(f _ 1(x) \\lt f _ 2(x)\\) \u304c\u3064\u306d\u306b\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u9577\u65b9\u5f62 \\(R\\) \uff1a \\(0 \\leqq x \\leqq 1 , \\ 0 \\leqq y \\leqq 2\\) \u3092\u8003\u3048\u308b. \u307e\u305f, \\(4\\) \u70b9 \\(\\text{P} _ 0 ( 0 , 1 )\\) , \\(\\text{P} _ 1 ( 0 , 0 )\\) , \\(\\text{P} _ 2 ( 1 , 1 )\\) , \\(\\text{P} _ 3 ( 1 , 0 )\\) \u3092\u3053\u306e\u9806\u306b\u7dda\u5206\u3067\u7d50\u3093\u3067\u5f97\u3089\u308c\u308b\u6298\u308c\u7dda\u3092 \\(L\\) \u3068\u3059\u308b.\r\n\u5b9f\u6570\u306e\u7d44 \\(( p , q )\\) \u3092, \u653e\u7269\u7dda \\(y = f(x)\\) \u3068\u6298\u308c\u7dda \\(L\\) \u306b\u5171\u6709\u70b9\u304c\u306a\u3044\u3088\u3046\u306a\u3059\u3079\u3066\u306e\u7d44\u306b\u308f\u305f\u3063\u3066\u52d5\u304b\u3059\u3068\u304d, \\(R\\) \u306e\u70b9\u306e\u3046\u3061\u3067\u653e\u7269\u7dda \\(y = f(x)\\) \u304c\u901a\u904e\u3059\u308b\u70b9\u5168\u4f53\u306e\u96c6\u5408\u3092 \\(T\\) \u3068\u3059\u308b.\r\n\\(R\\) \u304b\u3089 \\(T\\) \u3092\u9664\u3044\u305f\u9818\u57df \\(S\\) \u3092\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u56f3\u793a\u3057, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(y = f(x)\\) \u304c\u70b9 \\(( 0 , 1 )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nf(0) & = p^2+q = 1 \\\\\r\n\\text{\u2234} \\quad q & = 1-p^2 \\quad ... [1]\n\\end{align}\\]\r\n\u307e\u305f \\(y=f(x)\\) \u3068 \\(y=x\\) \u304c \\(x \\gt 0\\) \u306e\u90e8\u5206\u3067\u63a5\u3059\u308b\u306e\u3067, \\((x-p)^2+1-p^2 = x\\) , \u3059\u306a\u308f\u3061, \\(x^2-(2p+1)x+1 = 0 \\ ... [2]\\) \u304c\u6b63\u306e\u91cd\u89e3\u3092\u3082\u3066\u3070\u3088\u3044.
    \r\n[2] \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nD = (2p+1)^2 -4 \\cdot 1 & = 0 \\\\\r\n4p^2+4p-3 & = 0 \\\\\r\n(2p-1)(2p+3) & = 0 \\\\\r\n\\text{\u2234} \\quad p & = \\dfrac{1}{2} , -\\dfrac{3}{2}\n\\end{align}\\]\r\n\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a, \\(2p+1 \\gt 0\\) \u3064\u307e\u308a \\(p \\gt -\\dfrac{1}{2}\\) \u306a\u306e\u3067,\r\n\\[\r\np = \\dfrac{1}{2}\n\\]\r\n[1] \u3088\u308a\r\n\\[\r\nq = 1-\\left( \\dfrac{1}{2} \\right)^2 =\\dfrac{3}{4}\n\\]\r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u5b9f\u6570\u306e\u7d44\u306f\r\n\\[\r\n( p , q ) = \\underline{\\left( \\dfrac{1}{2} , \\dfrac{3}{4} \\right)}\n\\]\r\n\u3053\u306e\u3068\u304d\u306e\u63a5\u70b9\u306e\u5ea7\u6a19\u306f, [2] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nx^2-2x+1 & =0 \\\\\r\n\\text{\u2234} \\quad x & =1\n\\end{align}\\]\r\n\u63a5\u70b9\u306f \\(y = x\\) \u4e0a\u306b\u3042\u308b\u306e\u3067,\r\n\\[\r\ny=1\n\\]\r\n\u3086\u3048\u306b, \u63a5\u70b9\u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{( 1 , 1 )}\n\\]\r\n

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

    \\(g(x) = f _ 2(x) -f _ 1(x)\\) \u3068\u304a\u304f\u3068, \\(g(x)\\) \u306f \\(x\\) \u306b\u3064\u3044\u3066, \u9ad8\u3005 \\(1\\) \u6b21\u306e\u95a2\u6570\u3067\u3042\u308b.
    \r\n\u6761\u4ef6\u3088\u308a, \\(g( \\alpha ) \\gt 0 , \\ g( \\beta ) \\gt 0\\) \u306a\u306e\u3067, \\(\\alpha \\leqq x \\leqq \\beta\\) \u306b\u304a\u3044\u3066, \\(g(x) \\gt 0\\) , \u3059\u306a\u308f\u3061, \\(f _ 1(x) \\lt f _ 2(x)\\) .<\/p>\r\n

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

    (1)<\/strong> \u306e\u5834\u5408\u306e \\(f(x)\\) \u3092 \\(f _ 0(x) = \\left( x-\\dfrac{1}{2} \\right)^2 +\\dfrac{3}{4}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf _ 0(0) = 1 , \\ f _ 0(1) = 1 \\quad ... [3]\n\\]\r\n\\(f(x)\\) \u306b\u3064\u3044\u3066, \\(f(0) , f(1)\\) \u306e\u5024\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

    \"\"<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(0 \\leqq f(0) \\leqq 1\\) \u307e\u305f\u306f \\(0 \\leqq f(1) \\leqq 1\\) \u306e\u3068\u304d
      \r\n\\(y=f(x)\\) \u306f, \u7dda\u5206 \\(\\text{P} _ 0\\text{P} _ 1\\) \u307e\u305f\u306f \\(\\text{P} _ 2\\text{P} _ 3\\) \u3067 \\(L\\) \u3068\u5171\u6709\u70b9\u3092\u3082\u3064\u305f\u3081, \u4e0d\u9069.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\u300c \\(f(0) \\gt 1 , \\ f(1) \\lt 0\\) \u300d\u307e\u305f\u306f\u300c \\(f(0) \\lt 0 , \\ f(1) \\gt 1\\) \u300d\u306e\u3068\u304d
      \r\n\\(y=f(x)\\) \u306f, \u7dda\u5206 \\(\\text{P} _ 1\\text{P} _ 2\\) \u3067 \\(L\\) \u3068\u5171\u6709\u70b9\u3092\u3082\u3064\u305f\u3081, \u4e0d\u9069.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(f(0) \\lt 0 , \\ f(1) \\lt 0\\) \u306e\u3068\u304d
      \r\n\\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066, \\(y = f(x) \\lt 0\\) \u3067, \u9818\u57df \\(R\\) \u3092\u901a\u904e\u3057\u306a\u3044\u305f\u3081, \u4e0d\u9069.<\/p><\/li>\r\n

    4. 4*<\/strong>\u3000\\(f(0) \\gt 1 , \\ f(1) \\gt 1\\) \u306e\u3068\u304d
      \r\n(2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, [3] \u3088\u308a, \\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066, \\(f(x) \\gt f _ 0(x)\\) .
      \r\n\\(p = \\dfrac{1}{2}\\) \u3068\u56fa\u5b9a\u3057, \\(\\dfrac{3}{4} \\lt q \\leqq 2\\) \u3068\u52d5\u304b\u305b\u3070, \\(y = f(x)\\) \u306f\u9818\u57df \\(R\\) \u306e\u3046\u3061 \\(y = f _ 0(x)\\) \u3088\u308a\u4e0a\u5074\u3092\u3059\u3079\u3066\u901a\u904e\u3057, \u3053\u306e\u90e8\u5206\u304c\u9818\u57df \\(T\\) \u3068\u306a\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong>\uff5e4*<\/strong>\u3088\u308a, \u9818\u57df \\(S\\) \u306f, \u9818\u57df \\(R\\) \u306e\u3046\u3061 \\(y = f _ 0(x)\\) \u3088\u308a\u4e0b\u5074\u3067, \u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3092\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n

      \"\"<\/p>\r\n

      \u307e\u305f, \u3053\u306e\u90e8\u5206\u306e\u9762\u7a4d\u306f\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^1 f _ 0(x) \\, dx & = \\displaystyle\\int _ 0^1 (x^2-x+1) \\, dx \\\\\r\n& = \\left[ \\dfrac{x^3}{3} -\\dfrac{x^2}{2} +x \\right] _ 0^1 \\\\\r\n& = \\underline{\\dfrac{5}{6}}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570\u306e\u7d44 \\(( p , q )\\) \u306b\u5bfe\u3057, \\(f(x) = (x-p)^2 +q\\) \u3068\u304a\u304f. (1)\u3000\u653e\u7269\u7dda \\(y = f(x)\\) \u304c\u70b9 \\(( 0 , 1 )\\) \u3092\u901a\u308a, \u3057\u304b\u3082\u76f4\u7dda \\(y = 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":[32],"tags":[142,13],"class_list":["post-29","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2011","tag-osaka_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/29","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=29"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/29\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=29"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=29"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}