{"id":1933,"date":"2021-10-19T15:46:50","date_gmt":"2021-10-19T06:46:50","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1933"},"modified":"2021-10-19T15:46:50","modified_gmt":"2021-10-19T06:46:50","slug":"wsr201604","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201604\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52016\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(f(x) = x^3 -x\\) \u3068\u3059\u308b.\r\n\\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 \\(( p , q )\\) \u304b\u3089\u66f2\u7dda \\(y = f(x)\\) \u3078\u5f15\u3044\u305f\u63a5\u7dda\u3092\u8003\u3048\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u76f4\u7dda \\(y = m(x-p) +q\\) \u304c\u66f2\u7dda \\(y = f(x)\\) \u306e\u63a5\u7dda\u3068\u306a\u308b\u305f\u3081\u306e\u6761\u4ef6\u3092 \\(m , p , q\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 \\(( p , q )\\) \u304b\u3089\u66f2\u7dda \\(y = f(x)\\) \u306b \\(3\\) \u672c\u306e\u63a5\u7dda\u3092\u5f15\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u304d, \\(p , q\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u70b9 \\(( p , q )\\) \u306e\u7bc4\u56f2\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\r\nf'(x) = 3x^2 -1\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 \\(\\left( t , f(t) \\right)\\) \u306b\u304a\u3051\u308b \\(y = f(x)\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = ( 3 t^2 -1 ) ( x-t ) +t^3 -t \\\\\r\n& = ( 3 t^2 -1 ) x -2 t^3 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u306e\u50be\u304d\u304c \\(m\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n3 t^2 -1 & = m \\\\\r\n\\text{\u2234} \\quad t & = \\pm \\sqrt{\\dfrac{m+1}{3}} \\quad ... [2]\r\n\\end{align}\\]\r\n\u307e\u305f, \u70b9 \\(( p , q )\\) \u3092\u901a\u308b\u306e\u3067\r\n\\[\r\nq = ( 3 t^2 -1 ) p -2 t^3\r\n\\]\r\n\u3088\u3063\u3066, [2] \u3092\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\n\\underline{q = mp \\pm 2 \\left( \\dfrac{m+1}{3} \\right)^{\\frac{3}{2}}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n[1] \u3092\u6574\u7406\u3059\u308b\u3068\r\n\\[\r\n2 t^3 -3p t^2 +p +q = 0 \\quad ... [\\text{1'}]\r\n\\]\r\n\\(t\\) \u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f [1'] \u304c, \u7570\u306a\u308b \\(3\\) \u3064\u306e\u89e3\u3092\u6301\u3066\u3070\u3088\u3044.
    \r\n[1'] \u306e\u5de6\u8fba\u3092 \\(g(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ng'(t) = 6t ( t-p )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\np \\neq 0 \\quad ... [3] \\ \\text{\u304b\u3064} \\ f(0) f(p) \\lt 0 \\quad ... [4]\r\n\\]\r\n[4] \u306b\u3064\u3044\u3066\r\n\\[\r\n( p+q ) ( p^3 -p -q ) \\gt 0\r\n\\]\r\n\\(p = 0\\) \u306e\u3068\u304d, \\(( \\text{\u5de6\u8fba} ) = -q^2 \\leqq 0\\) \u306a\u306e\u3067, \u3053\u308c\u306f [3] \u3082\u307f\u305f\u3057\u3066\u3044\u308b.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{( p+q ) ( p^3 -p -q ) \\gt 0}\r\n\\]\r\n

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

    \\(g(p) = p^3 -p\\) \u3068\u304a\u3051\u3070\r\n\\[\r\ng'(p) = 3p^2 -1 = 3 \\left( p +\\dfrac{1}{\\sqrt{3}} \\right) \\left( p -\\dfrac{1}{\\sqrt{3}} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \\(g(p)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} p & \\cdots & -\\dfrac{1}{\\sqrt{3}} & \\cdots & \\dfrac{1}{\\sqrt{3}} & \\cdots \\\\ \\hline g'(p) & + & 0 & - & 0 & + \\\\ \\hline g(p) & \\nearrow & \\dfrac{2 \\sqrt{3}}{9} & \\searrow & -\\dfrac{2 \\sqrt{3}}{9} & \\nearrow \\end{array}\r\n\\]\r\n\\(g'(0) = -1\\) \u306a\u306e\u3067, \u539f\u70b9\u3067 \\(q = g(p)\\) \u3068 \\(q = -p\\) \u306f\u63a5\u3059\u308b.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u307e\u306a\u3044\uff09.<\/p>\r\n\"wsr20160401\"\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = x^3 -x\\) \u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 \\(( p , q )\\) \u304b\u3089\u66f2\u7dda \\(y = f(x)\\) \u3078\u5f15\u3044\u305f\u63a5\u7dda\u3092\u8003\u3048\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\u76f4\u7dda \\(y = m(x-p) + … \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":[157],"tags":[147,162],"class_list":["post-1933","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2016","tag-waseda_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1933","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=1933"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1933\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}