(1)<\/strong>\u3000\u6975\u5927\u5024\u304c\u6b63\u3067, \u6975\u5c0f\u5024\u304c\u8ca0\u3067, \u304b\u3064\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u306e\u548c\u304c\u8ca0\u3068\u306a\u308b\u70b9 \\((a,b)\\) \u306e\u7bc4\u56f2\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u6975\u5927\u5024\u304c \\(1\\) \u3067, \u6975\u5c0f\u5024\u304c \\(-1\\) \u3067\u3042\u308b\u3088\u3046\u306a\u70b9 \\((a,b)\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\r\nf'(x) = 3x^2+6ax+3b = 3 (x^2+2ax+b)\r\n\\]\r\n\\(f'(x) = 0\\) \u304c\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{gather}\r\n\\dfrac{D}{4} = a^2 -b \\gt 0 \\\\\r\n\\text{\u2234} \\quad b \\lt a^2 \\quad ... [1]\r\n\\end{gather}\\]\r\n\u3053\u306e\u3068\u304d, \\(2\\) \u3064\u306e\u89e3\u3092 \\(\\alpha , \\beta \\ ( \\alpha \\lt \\beta )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\alpha = -a -\\sqrt{a^2-b} & , \\ \\beta = -a+\\sqrt{a^2-b} \\\\\r\n\\text{\u2234} \\quad \\alpha +\\beta = -2a & , \\ \\alpha \\beta = b \\quad ... [2]\r\n\\end{align}\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nf( \\alpha ) f( \\beta ) & \\lt 0 \\quad ... [3] , \\\\\r\nf( \\alpha ) +f( \\beta ) & \\lt 0 \\quad ... [4]\r\n\\end{align}\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044. (2)<\/strong><\/p>\r\n \u6761\u4ef6\u3068 (1)<\/strong> \u306e\u9014\u4e2d\u7d4c\u904e\u3088\u308a\r\n\\[\\begin{align}\r\nf( \\alpha ) +f( \\beta ) & = 2a ( 2a^2-3b ) = 0 \\quad ... [7] , \\\\\r\nf( \\alpha ) f( \\beta ) & = b^2 ( 4b -3a^2 ) = -1\\quad ... [8]\r\n\\end{align}\\]\r\n[7] \u3092\u3068\u304f\u3068\r\n\\[\r\na=0 \\ \\text{\u307e\u305f\u306f} \\ a^2 = \\dfrac{3b}{2}\r\n\\]\r\n 1*<\/strong>\u3000\\(a = 0\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(a^2 = \\dfrac{3b}{2}\\) \u306e\u3068\u304d \u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n(a,b) = \\underline{\\left( 0 , -\\dfrac{1}{\\sqrt[3]{4}} \\right) , \\ \\left( \\pm \\dfrac{\\sqrt{3}}{\\sqrt[3]{2}} , \\sqrt[3]{2} \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b\\) \u306f\u5b9f\u6570\u3068\u3059\u308b. \u95a2\u6570 \\(f(x) = x^3+3ax^2+3bx\\) \u304c\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u3092\u3082\u3064. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6975\u5927\u5024\u304c\u6b63\u3067, \u6975\u5c0f\u5024\u304c\u8ca0\u3067, \u304b\u3064\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u306e\u548c\u304c\u8ca0\u3068\u306a\u308b\u70b9 \\(( … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u3053\u3053\u3067\r\n\\[\r\nf(x) = \\dfrac{x+a}{3} f'(x) -2(a^2-b)x-ab\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf( \\alpha ) & = -2(a^2-b) \\alpha -ab , \\\\\r\nf( \\beta ) & = -2(a^2-b) \\beta -ab\r\n\\end{align}\\]\r\n\u3053\u308c\u3068 [2] \u3092\u7528\u3044\u308c\u3070, [3] \u3088\u308a\r\n\\[\\begin{align}\r\nf( \\alpha ) f( \\beta ) = 4(a^2-b)^2 b -4a^2b(a^2-b) +a^2b^2 & \\lt 0 \\\\\r\n4a^4b -8a^2b^2 +4b^3 -4a^4b +4a^2b^2 +a^2b^2 & \\lt 0 \\\\\r\nb^2 ( 4b -3a^2 ) & \\lt 0 \\\\\r\n\\text{\u2234} \\quad b \\lt \\dfrac{3 a^2}{4} \\quad ... [5] &\r\n\\end{align}\\]\r\n\u307e\u305f, [4] \u3088\u308a\r\n\\[\\begin{gather}\r\nf( \\alpha ) +f( \\beta ) = 4a(a^2-b) -2ab \\lt 0 \\\\\r\na ( 2a^2-3b ) \\lt 0 \\\\\r\n\\text{\u2234} \\quad \\left\\{ \\begin{array}{ll} b \\lt \\dfrac{2a^2}{3} & \\ ( \\ a<0 \\text{\u306e\u3068\u304d} ) \\\\ b \\gt \\dfrac{2a^2}{3} & \\ ( \\ a>0 \\text{\u306e\u3068\u304d} ) \\end{array} \\right. \\quad ... [6]\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f, [1] \u304b\u3064 [5] \u304b\u3064 [6] \u306a\u306e\u3067, \u70b9 \\((a,b)\\) \u306e\u7bc4\u56f2\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u305f\u3060\u3057, \u25cb\u3068\u5883\u754c\u306f\u9664\u304f\uff09<\/p>\r\n![\"yokokoku_r_2007_04_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku_r_2007_04_01.png\")
\r\n\r\n
\r\n[8] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nb^3 & = -\\dfrac{1}{4} \\\\\r\n\\text{\u2234} \\quad b & = -\\dfrac{1}{\\sqrt[3]{4}}\r\n\\end{align}\\]<\/li>\r\n
\r\n[8] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\nb^2 \\left( 4b -3 \\cdot \\dfrac{3b}{2} \\right) = -1 \\\\\r\nb^3 = 2 \\\\\r\n\\text{\u2234} \\quad b = \\sqrt[3]{2}\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na = \\pm \\sqrt{\\dfrac{3 \\sqrt[3]{2}}{2}} = \\pm \\dfrac{\\sqrt{3}}{\\sqrt[3]{2}}\r\n\\]<\/li>\r\n<\/ol>\r\n