\\(f(x) = x^3-x\\) \u3068\u3057, \\(t\\) \u3092\u5b9f\u6570\u3068\u3059\u308b.\r\n\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u66f2\u7dda \\(y = f(x)\\) \u3092 \\(C _ 1\\) \u3068\u3057, \u76f4\u7dda \\(x=t\\) \u306b\u95a2\u3057\u3066 \\(C _ 1\\) \u3068\u5bfe\u79f0\u306a\u66f2\u7dda\r\n\\[\r\ny = f (2t-x)\r\n\\]\r\n\u3092 \\(C _ 2\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(C _ 1\\) \u3068 \\(C _ 2\\) \u304c \\(3\\) \u70b9\u3067\u4ea4\u308f\u308b\u3068\u304d, \\(t\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(t\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d \\(S\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf(2t-x) & = (2t-x)^3 -(2t-x) \\\\\r\n& = -x^3 +6tx^2 -( 12t^2-1 ) x +2t ( 4t^2-1 ) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(f(x) = f(2t-x)\\) \u3088\u308a\r\n\\[\\begin{align}\r\nx^3 -x = -x^3 +6tx^2 -( 12t^2-1) x & +2t(4t^2-1) \\\\\r\nx^3 -3tx^2 +( 6t^2 -1 )x -t( 4t^2-1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad (x-t) \\underline{ \\left\\{ x^2 -2tx +( 4t^2-1 ) \\right\\}} _ {[1]} & = 0 \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u304c \\(3\\) \u3064\u306e\u7570\u306a\u308b\u89e3\u3092\u3082\u3064\u306e\u306f, \\([1] = 0\\) \u304c \\(x = t\\) \u4ee5\u5916\u306e\u7570\u306a\u308b \\(2\\) \u3064\u306e\u89e3\u3092\u3082\u3064\u3068\u304d\u3067\u3042\u308b. (2)<\/strong><\/p>\r\n \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = 2t , \\ \\alpha \\beta = 4t^2-1 \\quad ... [2] \\ .\r\n\\]\r\n\u56f2\u307e\u308c\u305f\u90e8\u5206\u3082 \\(x=t\\) \u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS & = 2 \\displaystyle\\int _ {\\alpha}^{t} \\left\\{ f(x) -f(2t-x) \\right\\} \\, dx \\\\\r\n& = 4 \\displaystyle\\int _ {\\alpha}^{t} (x-t)( x-\\alpha )( x-\\beta ) \\, dx \\\\\r\n& = 4 \\displaystyle\\int _ {\\alpha}^{t} (x-t)( x-t -\\alpha +t )( x -t -\\beta +t ) \\, dx \\\\\r\n& = 4 \\displaystyle\\int _ {\\alpha}^{t} \\left\\{ (x-t)^3 -\\left( \\alpha +\\beta -2t \\right) (x-t)^2 \\right. \\\\\r\n& \\qquad \\qquad \\qquad \\left. +\\left( \\alpha -t \\right) \\left( \\beta -t \\right) (x-t) \\right\\} \\, dx \\\\\r\n& = 4 \\left[ \\dfrac{(x-t)^4}{4} -(1-3t^2) \\dfrac{(x-t)^2}{2} \\right] _ {\\alpha}^{t} \\\\\r\n& = -(1-3t^2)^2 +2 (1-3t^2)^2 \\\\\r\n& = (1-3t^2)^2 \\leqq 1 \\ .\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f \\(t = 0\\) \u306e\u3068\u304d\u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\([1] = 0\\) \u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} = t^2 -( 4t^2-1 ) & \\gt 0 \\\\\r\n3t^2 -1 & \\lt 0 \\\\\r\n\\text{\u2234} \\quad -\\dfrac{\\sqrt{3}}{3} \\lt t & \\lt \\dfrac{\\sqrt{3}}{3} \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \\(2\\) \u3064\u306e\u89e3\u3092 \\(\\alpha , \\beta \\ ( \\alpha \\lt \\beta )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\alpha = t - \\sqrt{1 -3t^2} , \\ \\beta = t + \\sqrt{1 -3t^2} \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(\\alpha , \\beta \\neq t\\) \u3067\u3042\u308b.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{-\\dfrac{\\sqrt{3}}{3} \\lt t \\lt \\dfrac{\\sqrt{3}}{3}} \\ .\r\n\\]\r\n
\r\n\u3088\u3063\u3066, \\(S\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{1} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(f(x) = x^3-x\\) \u3068\u3057, \\(t\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u66f2\u7dda \\(y = f(x)\\) \u3092 \\(C _ 1\\) \u3068\u3057, \u76f4\u7dda \\(x=t\\) \u306b\u95a2\u3057\u3066 \\(C _ 1\\) \u3068\u5bfe … \u7d9a\u304d\u3092\u8aad\u3080