\u70b9 \\(P\\) \u304b\u3089\u653e\u7269\u7dda \\(y=\\dfrac{1}{2} x^2\\) \u3078 \\(2\\) \u672c\u306e\u63a5\u7dda\u304c\u5f15\u3051\u308b\u3068\u304d, \\(2\\) \u3064\u306e\u63a5\u70b9\u3092 \\(A , B\\) \u3068\u3057, \u7dda\u5206 \\(PA , PB\\) \u304a\u3088\u3073\u3053\u306e\u653e\u7269\u7dda\u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S\\) \u3068\u3059\u308b. \\(PA , PB\\) \u304c\u76f4\u4ea4\u3059\u308b\u3068\u304d\u306e \\(S\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(y=\\dfrac{1}{2}x^2\\) \u3088\u308a, \\(y'=x\\) .
\r\n\\(A\\) \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(a \\ ( a \\gt 0 )\\) \u3068\u304a\u304f\u3068, \\(AP\\) \u306e\u50be\u304d\u306f \\(a\\) \u306a\u306e\u3067, \\(BP\\) \u306e\u50be\u304d\u306f \\(-\\dfrac{1}{a}\\) \u3067\u3042\u308a, \\(B\\) \u306e \\(x\\) \u5ea7\u6a19\u306f \\(-\\dfrac{1}{a}\\) \u3068\u306a\u308b.
\r\n\\(PA , PB\\) \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\nPA : \\ y & = a(x-a) +\\dfrac{a^2}{2} = ax +\\dfrac{a^2}{2} \\\\\r\nPB : \\ y & = -\\dfrac{1}{a} \\left( x+\\dfrac{1}{a} \\right) +\\dfrac{1}{2a^2} = -\\dfrac{x}{a} +\\dfrac{1}{2a^2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066,\u4ea4\u70b9 \\(P\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nax +\\dfrac{a^2}{2} & = -\\dfrac{x}{a} +\\dfrac{1}{2a^2} \\\\\r\n\\left( a+\\dfrac{1}{a} \\right) x & = \\dfrac{1}{2} \\left( a^2+\\dfrac{1}{a^2} \\right) \\\\\r\n\\text{\u2234} \\quad x & =\\dfrac{1}{2} \\left( a-\\dfrac{1}{a} \\right) \\\\\r\n\\text{\u2234} \\quad y & = \\dfrac{a}{2} \\left( a-\\dfrac{1}{a} \\right) +\\dfrac{a^2}{2} = -\\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nP \\ \\left( \\dfrac{1}{2} \\left( a-\\dfrac{1}{a} \\right) , -\\dfrac{1}{2} \\right)\r\n\\]\r\n\\(AB\\) \u306e\u4e2d\u70b9\u3092 \\(M\\) \u3068\u304a\u304f\u3068, \u305d\u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( \\dfrac{a-\\frac{1}{a}}{2} , \\dfrac{\\frac{a^2}{2} +\\frac{1}{2a^2}}{2} \\right)\r\n\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\left( \\dfrac{1}{2} \\left( a-\\dfrac{1}{a} \\right) , \\dfrac{1}{4} \\left( a^2+\\dfrac{1}{a^2} \\right) \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f, \\(\\triangle ABP\\) \u306e\u9762\u7a4d\u3092 \\(S _ 1 , C\\) \u3068 \\(AB\\) \u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S _ 2\\) \u3068\u304a\u3051\u3070,\r\n\\[\r\nS = S _ 1 -S _ 2\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nS _ 1 & = \\dfrac{1}{2} \\left( a+\\dfrac{1}{a} \\right) \\left\\{ \\dfrac{1}{4} \\left( a^2+\\dfrac{1}{a^2} \\right) +\\dfrac{1}{2} \\right\\} \\\\\r\n& = \\dfrac{1}{8} \\left( a+\\dfrac{1}{a} \\right)^3 , \\\\\r\nS _ 2 & = \\dfrac{1}{2} \\cdot \\dfrac{1}{6} \\left( a+\\dfrac{1}{a} \\right)^3 \\\\\r\n& = \\dfrac{1}{12} \\left( a+\\dfrac{1}{a} \\right)^3\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u306e\u95a2\u4fc2\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{24} \\left( a+\\dfrac{1}{a} \\right)^3 \\\\\r\n& \\geqq \\dfrac{1}{24} \\left( 2 \\sqrt{ a \\cdot \\dfrac{1}{a}} \\right)^3 = \\dfrac{1}{3}\r\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f, \\(a =\\dfrac{1}{a}\\) \u3059\u306a\u308f\u3061 \\(a=1\\) \u306e\u3068\u304d.
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\dfrac{1}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u70b9 \\(P\\) \u304b\u3089\u653e\u7269\u7dda \\(y=\\dfrac{1}{2} x^2\\) \u3078 \\(2\\) \u672c\u306e\u63a5\u7dda\u304c\u5f15\u3051\u308b\u3068\u304d, \\(2\\) \u3064\u306e\u63a5\u70b9\u3092 \\(A , B\\) \u3068\u3057, \u7dda\u5206 \\(PA , PB\\) \u304a\u3088\u3073\u3053\u306e\u653e\u7269\u7dda\u3067\u56f2\u307e\u308c … \u7d9a\u304d\u3092\u8aad\u3080