{"id":158,"date":"2011-12-01T18:09:47","date_gmt":"2011-12-01T09:09:47","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=158"},"modified":"2021-11-04T09:41:51","modified_gmt":"2021-11-04T00:41:51","slug":"htb201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201001\/","title":{"rendered":"\u4e00\u6a4b\u59272010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\u5b9f\u6570 \\(p , q , r\\) \u306b\u5bfe\u3057\u3066, \\(3\\) \u6b21\u65b9\u7a0b\u5f0f \\(f(x)\\) \u3092 \\(f(x) = x^3 +px^2 +qx +r\\) \u3068\u5b9a\u3081\u308b.\r\n\u5b9f\u6570 \\(a , c\\) \u304a\u3088\u3073 \\(0\\) \u3067\u306a\u3044\u5b9f\u6570 \\(b\\) \u306b\u5bfe\u3057\u3066, \\(a +bi\\) \u3068 \\(c\\) \u306f\u3044\u305a\u308c\u3082\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306e\u89e3\u3067\u3042\u308b\u3068\u3059\u308b. \u305f\u3060\u3057 \\(i\\) \u306f\u865a\u6570\u5358\u4f4d\u3092\u8868\u3059.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306b\u304a\u3044\u3066, \u70b9 \\(\\left( a , f(a) \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u50be\u304d\u3092 \\(s(a)\\) \u3068\u3057, \u70b9 \\(\\left( c , f(c) \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u50be\u304d\u3092 \\(s(c)\\) \u3068\u3059\u308b. \\(a \\neq c\\) \u306e\u3068\u304d, \\(s(a)\\) \u3068 \\(s(c)\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u305b\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3055\u3089\u306b, \\(a , c\\) \u306f\u6574\u6570\u3067\u3042\u308a, \\(b\\) \u306f \\(0\\) \u3067\u306a\u3044\u6574\u6570\u3067\u3042\u308b\u3068\u3059\u308b. \u6b21\u3092\u8a3c\u660e\u305b\u3088.\r\n

      \r\n

    1. (i)<\/strong>\u3000\\(p , q , r\\) \u306f\u3059\u3079\u3066\u6574\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n

    2. (ii)<\/strong>\u3000\\(p\\) \u304c \\(2\\) \u306e\u500d\u6570\u3067\u3042\u308a, \\(q\\) \u304c \\(4\\) \u306e\u500d\u6570\u3067\u3042\u308b\u306a\u3089\u3070, \\(a , b , c\\) \u306f\u3059\u3079\u3066 \\(2\\) \u306e\u500d\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol><\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

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

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

      \\(a+bi\\) \u3068\u5171\u5f79\u306a \\(a-bi\\) \u3082 \\(f(x) = 0\\) \u306e\u89e3\u3067\u3042\u308b.
      \r\n\u3057\u305f\u304c\u3063\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\\begin{align}\r\np & = -( a+bi ) -( a-bi ) -c = -( 2a+c ) \\quad ... [1] , \\\\\r\nq & = ( a+bi )( a-bi ) + ( a+bi ) c +( a-bi ) c \\\\\r\n& = a^2 +b^2 +2ac \\quad ... [2] , \\\\\r\nr & = -( a+bi )( a-bi ) c = -( a^2 +b^2 ) c \\quad ... [3]\r\n\\end{align}\\]\r\n\u3068\u3053\u308d\u3067, \\(s(x) = f'(x) = 3x^2 +2px +q\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\ns(c) -s(a) & = 3 \\left( c^2 -a^2 \\right) +2p ( c-a ) \\\\\r\n& = ( c-a ) \\{ 3c +3a -4a -2c \\} \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = ( c-a )^2 \\gt 0 \\quad ( \\ \\text{\u2235} \\ a \\neq c \\ )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{s(c) \\gt s(a)}\r\n\\]\r\n

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

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

      \\(a , b , c\\) \u304c\u6574\u6570\u306a\u3089\u3070, \\(2a+c\\) , \\(a^2+b^2+2ac\\) , \\(( a^2+b^2 )c\\) \u3082\u6574\u6570\u306a\u306e\u3067, [1]\uff5e[3] \u3088\u308a \\(p , q , r\\) \u306f\u6574\u6570\u3067\u3042\u308b.<\/p>\r\n

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

      \u6761\u4ef6\u3088\u308a, \\(p = 2p'\\) , \\(q = 4q'\\) \uff08 \\(p' , q'\\) \u306f\u6574\u6570\uff09\u3068\u304a\u3051\u308b.
      \r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\n2p' & = -( 2a+c ) \\\\\r\n\\text{\u2234} \\quad c & = -2( a+p' )\r\n\\end{align}\\]\r\n\\(a+p'\\) \u306f\u6574\u6570\u306a\u306e\u3067, \\(c\\) \u306f \\(2\\) \u306e\u500d\u6570\u3067\u3042\u308b\u304b\u3089, \\(c = 2c'\\) \uff08 \\(c'\\) \u306f\u6574\u6570\uff09\u3068\u304a\u3051\u308b.
      \r\n[2] \u3088\u308a\r\n\\[\\begin{align}\r\n4q' = a^2 +b^2 & +4ac' \\\\\r\n\\text{\u2234} \\quad a^2 +b^2 & = 4( q'-ac' )\r\n\\end{align}\\]\r\n\\(q'-ac'\\) \u306f\u6574\u6570\u306a\u306e\u3067, \\(a^2 +b^2\\) \u306f \\(4\\) \u306e\u500d\u6570.
      \r\n\u6574\u6570 \\(m\\) \u306b\u3064\u3044\u3066<\/p>\r\n