- \r\n
- (\uff0a)<\/strong>\u3000\\(\\ell\\) \u306e\u50be\u304d\u306f \\(1\\) \u3067, \u66f2\u7dda \\(y = x^3 -2x\\) \u3068\u7570\u306a\u308b \\(3\\) \u70b9\u3067\u4ea4\u308f\u308b.<\/li>\r\n<\/ol>\r\n
\u305d\u306e\u4ea4\u70b9\u3092 \\(x\\) \u5ea7\u6a19\u304c\u5c0f\u3055\u306a\u3082\u306e\u304b\u3089\u9806\u306b P, Q, R \u3068\u3057, \u3055\u3089\u306b\u7dda\u5206 PQ \u306e\u4e2d\u70b9\u3092 S \u3068\u3059\u308b.<\/p>\r\n
- \r\n
(1)<\/strong>\u3000\u70b9 R \u306e\u5ea7\u6a19\u3092 \\(( a , a^3 -2x )\\) \u3068\u3059\u308b\u3068\u304d, \u70b9 S \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n
(2)<\/strong>\u3000\u76f4\u7dda \\(\\ell\\) \u304c\u6761\u4ef6 (\uff0a)<\/strong> \u3092\u6e80\u305f\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u304d, \u70b9 S \u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n
(3)<\/strong>\u3000\u76f4\u7dda \\(\\ell\\) \u304c\u6761\u4ef6 (\uff0a)<\/strong> \u3092\u6e80\u305f\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u304d, \u7dda\u5206 PS \u304c\u52d5\u3044\u3066\u3067\u304d\u308b\u9818\u57df\u306e\u9762\u7a4d\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
\\(\\ell : \\ y = x+k\\) \u3068\u304a\u304f\u3068, \u66f2\u7dda\u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3057\u3066\r\n\\[\\begin{align}\r\nx^3 -2x & = x +k \\\\\r\n\\text{\u2234} \\quad x^3 -3x -k & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\nP, Q \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p , q\\) \u3068\u304a\u3051\u3070, \u3053\u308c\u3089\u3068 \\(a\\) \u304c [1] \u306e \\(3\\) \u89e3\u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\left\\{ \\begin{array}{ll} p +q +a = 0 & ... [2] \\\\ pq +a ( p+q ) = -3& ... [3] \\\\ apq = k & ... [4] \\end{array} \\right.\r\n\\]\r\n[2] \u3088\u308a, \\(p+q = -a\\) .
\r\n[3] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\npq -a^2 & = -3 \\\\\r\n\\text{\u2234} \\quad pq & = a^2 -3\r\n\\end{align}\\]\r\n[4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\r\nk = a ( a^2 -3 )\r\n\\]\r\n\u70b9 S \u306e \\(x\\) \u5ea7\u6a19\u306f \\(\\dfrac{p+q}{2} = -\\dfrac{a}{2}\\) \u3067, \\(\\ell\\) \u4e0a\u306e\u70b9\u306a\u306e\u3067, \\(y\\) \u5ea7\u6a19\u306f\r\n\\[\r\n-\\dfrac{a}{2} +a ( a^2 -3 ) = a^3 -\\dfrac{7a}{2}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( -\\dfrac{a}{2} , a^3 -\\dfrac{7a}{2} \\right)}\r\n\\]\r\n(2)<\/strong><\/p>\r\n[1] \u306e\u5de6\u8fba\u3092 \\(f(x)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(x) = 3x^2 -3 = 3 (x+1) (x-1)\r\n\\]\r\n\u3086\u3048\u306b, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} x & \\cdots & -1 & \\cdots & 1 & \\cdots \\\\ \\hline f'(x) & + & 0 & - & 0 & + \\\\ \\hline f(x) & \\nearrow & 2-k & \\searrow & -2-k & \\nearrow \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u304c\u7570\u306a\u308b \\(3\\) \u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n2-k \\gt 0 \\ & \\text{\u304b\u3064} \\ -2-k \\lt 0 \\\\\r\n-2 & \\lt a ( a^2 -3 ) \\lt 2 \\\\\r\na^3 -3a +2 \\gt 0 \\ & \\text{\u304b\u3064} \\ a^3 -3a -2 \\lt 0 \\\\\r\n(a-1) ( a^2 +a +2 ) \\gt 0 \\ & \\text{\u304b\u3064} \\ (a+1)^2 (a-2) \\lt 0\r\n\\end{align}\\]\r\n\\(a^2 +a +2 = \\left( a +\\dfrac{1}{2} \\right)^2 +\\dfrac{7}{4} \\gt 0\\) , \\((a+1)^2 \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\na-1 \\gt 0 \\ & \\text{\u304b\u3064} \\ a-2 \\lt 0 \\\\\r\n\\text{\u2234} \\quad & 1 \\lt a \\lt 2 \\quad ... [5]\r\n\\end{align}\\]\r\n\u70b9 S \\(( X , Y )\\) \u3068\u304a\u3051\u3070, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(X = -\\dfrac{a}{2}\\) \u306a\u306e\u3067\r\n\\[\r\na = -2X\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\nY = (-2X)^3 +\\dfrac{7}{2} \\cdot 2X = -8 X^3 +7X\r\n\\]\r\n\u307e\u305f, [5] \u3088\u308a\r\n\\[\\begin{align}\r\n1 & \\lt -2X \\lt 2 \\\\\r\n\\text{\u2234} \\quad -1 & \\lt X \\lt -\\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u66f2\u7dda} \\ : \\ y = -8x^3 +7x \\ \\left( -1 \\lt x \\lt -\\dfrac{1}{2} \\right)}\r\n\\]\r\n
(3)<\/strong><\/p>\r\n[5] \u3088\u308a \\(\\ell\\) \u306f \\(y = x-2\\) \u304b\u3089 \\(y = x+2\\) \u307e\u3067\u52d5\u304d,
\r\n\u70b9 S \u306e \\(x\\) \u5ea7\u6a19\u306f \\(-\\dfrac{1}{2}\\) \u304b\u3089 \\(-1\\) \u306b\u5909\u5316\u3059\u308b.
\r\n\u70b9 S \u306e\u8ecc\u8de1\u306e\u5f0f\u3068 \\(y = x-2\\) \u304b\u3089\r\n\\[\\begin{align}\r\n-8x^3 +7x & = x-2 \\\\\r\n4x^3 -6x -1 & = 0 \\\\\r\n( 2x +1 )^2 (x-1) & = 0 \\\\\r\n\\text{\u2234} \\quad x & = -\\dfrac{1}{2} , 1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u7dda\u5206 PS \u306e\u52d5\u304f\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8.<\/p>\r\n![\"thr20210401\"](\"\/nyushi\/wp-content\/uploads\/thr20210401.svg\")
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {-2}^{-1} \\left\\{ ( x^3 -2x ) -(x-2) \\right\\} \\, dx \\\\\r\n& \\qquad +\\displaystyle\\int _ {-1}^{-\\frac{1}{2}} \\left\\{ ( -8x^3 +7x ) -(x-2) \\right\\} \\, dx \\\\\r\n& = \\displaystyle\\int _ {-2}^{-1} ( x^3 -3x +2 ) \\, dx +2 \\displaystyle\\int _ {-1}^{-\\frac{1}{2}} ( -4x^3 +3x +1 ) \\, dx \\\\\r\n& = \\left[ \\dfrac{x^4}{4} -\\dfrac{3}{2} x^2 +2x \\right] _ {-2}^{-1} +2 \\left[ -x^4 +\\dfrac{3}{2} x^2 +x \\right] _ {-1}^{-\\frac{1}{2}} \\\\\r\n& = -\\dfrac{13}{4} +6 -\\dfrac{3}{8} +1 \\\\\r\n& = \\underline{\\dfrac{27}{8}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3059\u76f4\u7dda \\(\\ell\\) \u3092\u8003\u3048\u308b. (\uff0a)\u3000\\(\\ell\\) \u306e\u50be\u304d\u306f \\(1\\) \u3067, \u66f2\u7dda \\(y = x^3 -2x\\) \u3068\u7570\u306a\u308b \\(3\\) \u70b9\u3067\u4ea4\u308f\u308b. \u305d\u306e\u4ea4\u70b9\u3092 … \u7d9a\u304d\u3092\u8aad\u3080