{"id":1985,"date":"2021-11-21T15:55:54","date_gmt":"2021-11-21T06:55:54","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1985"},"modified":"2021-11-23T09:55:44","modified_gmt":"2021-11-23T00:55:44","slug":"osr202101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr202101\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2021\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(a , b\\) \u3092 \\(ab \\lt 1\\) \u3092\u307f\u305f\u3059\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b.\r\n\\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 P \\(( a , b )\\) \u304b\u3089, \u66f2\u7dda \\(y = \\dfrac{1}{x} \\ ( x \\gt 0 )\\) \u306b \\(2\\) \u672c\u306e\u63a5\u7dda\u3092\u5f15\u304d,\r\n\u305d\u306e\u63a5\u70b9\u3092 Q \\(\\left( s , \\dfrac{1}{s} \\right)\\) , R \\(\\left( t , \\dfrac{1}{t} \\right)\\) \u3068\u3059\u308b.
\r\n\u305f\u3060\u3057, \\(s \\lt t\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(s\\) \u304a\u3088\u3073 \\(t\\) \u3092 \\(a , b\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 P \\(( a , b)\\) \u304c\u66f2\u7dda \\(y = \\dfrac{9}{4} -3 x^2\\) \u4e0a\u306e \\(x \\gt 0\\) , \\(y \\gt 0\\) \u3092\u307f\u305f\u3059\u90e8\u5206\u3092\u52d5\u304f\u3068\u304d, \\(\\dfrac{t}{s}\\) \u306e\u6700\u5c0f\u5024\u3068\u305d\u306e\u3068\u304d\u306e \\(a , b\\) \u306e\u5024\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 = \\dfrac{1}{x}\\) \u3088\u308a, \\(y' = -\\dfrac{1}{x^2}\\) \u306a\u306e\u3067, \u70b9 \\(\\left( p , \\dfrac{1}{p} \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = -\\dfrac{1}{p^2} (x-p) +\\dfrac{1}{p} \\\\\r\n& = -\\dfrac{x}{p^2} +\\dfrac{2}{p}\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u70b9 P \u3092\u901a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nb = -\\dfrac{a}{p^2} +\\dfrac{2}{p} & \\\\\r\n\\text{\u2234} \\quad b p^2 -2p +a & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n[1] \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\dfrac{D}{4} = 1 -ab \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306e \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u304c \\(s , t\\) \u3067\u3042\u308b.
    \r\n\u3088\u3063\u3066, \\(s \\lt t\\) \u306a\u306e\u3067\r\n\\[\r\ns = \\underline{\\dfrac{1 -\\sqrt{1 -ab}}{b}} \\ , \\ t = \\underline{\\dfrac{1 +\\sqrt{1 -ab}}{b}}\r\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{t}{s} & = \\dfrac{1 +\\sqrt{1 -ab}}{1 -\\sqrt{1 -ab}} \\\\\r\n& = -1 +\\dfrac{2}{1 -\\sqrt{1 -ab}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\dfrac{t}{s}\\) \u304c\u6700\u5c0f\u3068\u306a\u308b\u306e\u306f, \\(ab\\) \u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d.
    \r\n\u6761\u4ef6\u3088\u308a, \\(a\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f \\(0 \\lt a \\lt \\dfrac{\\sqrt{3}}{2}\\) \u3067\u3042\u308a\r\n\\[\r\nab = a \\left( \\dfrac{9}{4} -3 a^2 \\right) = \\dfrac{9a}{4} -3 a^3\r\n\\]\r\n\u3053\u308c\u3092 \\(f(a)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(a) = \\dfrac{9}{4} -9a^2 = \\dfrac{9}{4} ( 1 -4a^2 )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} a & (0) & \\cdots & \\dfrac{1}{2} & \\cdots & \\left( \\dfrac{\\sqrt{3}}{2} \\right) \\\\ \\hline f'(a) & & + & 0 & - & \\\\ \\hline f(a) & (0) & \\nearrow & \\dfrac{3}{4} & \\searrow & (0) \\end{array}\r\n\\]\r\n\\(ab = \\dfrac{3}{4} \\lt 1\\) \u306a\u306e\u3067, \\(s , t\\) \u3082\u5b58\u5728\u3059\u308b.
    \r\n\u3088\u3063\u3066, \\(a = \\underline{\\dfrac{1}{2}}\\) , \\(b = \\underline{\\dfrac{3}{2}}\\) \u306e\u3068\u304d, \\(\\dfrac{t}{s}\\) \u306e\u6700\u5c0f\u5024\u306f\r\n\\[\r\n-1 +\\dfrac{2}{1 -\\sqrt{\\dfrac{1}{4}}} = \\underline{3}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b\\) \u3092 \\(ab \\lt 1\\) \u3092\u307f\u305f\u3059\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 P \\(( a , b )\\) \u304b\u3089, \u66f2\u7dda \\(y = \\dfrac{1}{x} \\ ( x \\gt 0 )\\) \u306b … \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":[172],"tags":[142,165],"class_list":["post-1985","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2021","tag-osaka_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1985","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=1985"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1985\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}