\\(xy\\) \u5e73\u9762\u4e0a\u306b\u653e\u7269\u7dda \\(C : \\ y = -3x^2 +3\\) \u3068 \\(2\\) \u70b9 A \\(( 1 , 0 )\\) , P \\(( 0 , 3p )\\) \u304c\u3042\u308b.\r\n\u7dda\u5206 AP \u3068 \\(C\\) \u306f, A \u3068\u306f\u7570\u306a\u308b\u70b9 Q \u3092\u5171\u6709\u3057\u3066\u3044\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u5b9a\u6570 \\(p\\) \u306e\u5b58\u5728\u3059\u308b\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(\\text{S} _ 1\\) \u3092, \\(C\\) \u3068\u7dda\u5206 AQ \u3067\u56f2\u307e\u308c\u305f\u9818\u57df\u3068\u3057, \\(\\text{S} _ 2\\) \u3092, \\(C\\) , \u7dda\u5206 QP , \u304a\u3088\u3073 \\(y\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u9818\u57df\u3068\u3059\u308b. \\(\\text{S} _ 1\\) \u3068 \\(\\text{S} _ 2\\) \u306e\u9762\u7a4d\u306e\u548c\u304c\u6700\u5c0f\u306b\u306a\u308b \\(p\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\r\n\\text{\u7dda\u5206 AP} : \\ y = -3px+3p \\quad ( 0 \\leqq x \\leqq 1 )\r\n\\]\r\n\u3053\u308c\u3068 \\(C : \\ y = -3x^2+3\\) \u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u306f\r\n\\[\\begin{align}\r\n-3px+3p & = -3x^2+3 \\\\\r\nx^2 -px +p-1 & = 0 \\\\\r\n( x-1 )( x-p+1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad x = 1 , p-1 &\r\n\\end{align}\\]\r\n\\(0 \\leqq x \\leqq 1\\) \u306b\u304a\u3044\u3066, \\(2\\) \u3064\u306e\u4ea4\u70b9\u3092\u6301\u3064\u306e\u3067\r\n\\[\\begin{gather}\r\n0 \\leqq p-1 \\lt 1 \\\\\r\n\\text{\u2234} \\quad \\underline{1 \\leqq p \\lt 2}\r\n\\end{gather}\\]\r\n (2)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\text{S} _ 1 +\\text{S} _ 2 & = \\dfrac{1}{2} \\cdot 1 \\cdot 3p -\\dfrac{1}{2} \\displaystyle\\int _ {-1}^1 (-3x^2+3) \\, dx \\\\\r\n& \\qquad +2 \\displaystyle\\int _ {p-1}^1 \\left\\{ (-3x^2+3) -(-3px+3p) \\right\\} \\, dx \\\\\r\n& = \\dfrac{3p}{2} -\\dfrac{1}{2} \\cdot \\dfrac{3}{6} \\left\\{ 1 -(-1) \\right\\}^3 +2 \\cdot \\dfrac{3}{6} \\left\\{ 1 -(p-1) \\right\\}^3 \\\\\r\n& = \\dfrac{3p}{2} -2 +(2-p)^3 \\\\\r\n& = \\dfrac{1}{2} ( \\underline{-2p^3+12p^2 -21p +12} )\r\n\\end{align}\\]\r\n\u3053\u306e\u4e0b\u7dda\u90e8\u3092 \\(f(p)\\) \u3068\u304a\u304f\u3068,\r\n\\[\\begin{align}\r\nf'(p) & = -6p^2+24p-21 \\\\\r\n& =-6\\left( p-\\dfrac{4+\\sqrt{2}}{2} \\right) \\left( p-\\dfrac{4-\\sqrt{2}}{2} \\right)\r\n\\end{align}\\]\r\n\\(1 \\leqq p \\lt 2\\) \u306b\u304a\u3044\u3066, \\(f'(p) = 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\np = \\dfrac{4-\\sqrt{2}}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(p)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} p & 1 & \\cdots & \\dfrac{4-\\sqrt{2}}{2} & \\cdots & (2) \\\\ \\hline\r\nf'(p) & & - & 0 & + & \\\\ \\hline f(p) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(p\\) \u306e\u5024\u306f\r\n\\[\r\np = \\underline{\\dfrac{4-\\sqrt{2}}{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306b\u653e\u7269\u7dda \\(C : \\ y = -3x^2 +3\\) \u3068 \\(2\\) \u70b9 A \\(( 1 , 0 )\\) , P \\(( 0 , 3p )\\) \u304c\u3042\u308b. \u7dda\u5206 AP \u3068 \\(C\\) \u306f, A \u3068\u306f\u7570 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n