\u66f2\u7dda \\(y = \\dfrac{1}{2} ( x^2 +1 )\\) \u4e0a\u306e\u70b9 P \u306b\u304a\u3051\u308b\u63a5\u7dda\u306f \\(x\\) \u8ef8\u3068\u4ea4\u308f\u308b\u3068\u3057,\r\n\u305d\u306e\u4ea4\u70b9\u3092 Q \u3068\u304a\u304f. \u7dda\u5206 PQ \u306e\u9577\u3055\u3092 \\(L\\) \u3068\u3059\u308b\u3068\u304d,\r\n\\(L\\) \u304c\u53d6\u308a\u3046\u308b\u5024\u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p>\r\n
P \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(t\\) \u3068\u304a\u304f.
\r\n\u66f2\u7dda \\(y = \\dfrac{1}{2} ( x^2 +1 )\\) \u306f \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067, \\(t=0\\) \u306e\u3068\u304d\u306b\u63a5\u7dda\u306f \\(x\\) \u8ef8\u3068\u4ea4\u308f\u3089\u306a\u3044\u306e\u3067, \\(t \\gt 0\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
\r\n\\(y' = x\\) \u306a\u306e\u3067, PQ \u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = t ( x-t ) +\\dfrac{1}{2} ( t^2 +1 ) \\\\\r\n& = tx -\\dfrac{1}{2} ( t^2 -1 )\r\n\\end{align}\\]\r\n\\(y = 0\\) \u3068\u3059\u308c\u3070\r\n\\[\\begin{align}\r\ntx & = \\dfrac{1}{2} ( t^2 -1 ) \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{t^2 -1}{2t}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, Q \\(\\left( \\dfrac{t^2 -1}{2t} , 0 \\right)\\) \u3067\u3042\u308a\r\n\\[\\begin{align}\r\nL^2 & = \\left( t -\\dfrac{t^2 -1}{2t} \\right)^2 +\\left( \\dfrac{t^2 +1}{2} \\right)^2 \\\\\r\n& = \\dfrac{( t^2 +1 )^2}{4 t^2} +\\dfrac{( t^2 +1 )^2}{4} \\\\\r\n& = \\dfrac{1}{4} \\cdot \\underline{\\dfrac{( t^2 +1 )^3}{t^2}} _ {[1]}\r\n\\end{align}\\]\r\n[1] \u306b\u3064\u3044\u3066, \\(s = t^2 ( \\gt 0 )\\) \u3068\u304a\u3044\u3066 \\(f(s) = \\dfrac{(s+1)^3}{s}\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nf'(s) & = \\dfrac{3 (s+1)^2 s -(s+1)^3}{s^2} \\\\\r\n& = \\dfrac{(s+1)^2 ( 2s -1 )}{s^2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(s)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} s & (0) & \\cdots & \\dfrac{1}{2} & \\cdots \\\\ \\hline f'(s) & & - & 0 & + \\\\ \\hline f(s) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(L\\) \u306f \\(s = \\dfrac{1}{2}\\) \u306e\u3068\u304d\u306b\u6700\u5c0f\u3068\u306a\u308a, \u305d\u306e\u5024\u306f\r\n\\[\r\n\\sqrt{\\dfrac{1}{4} f \\left( \\dfrac{1}{2} \\right)} = \\sqrt{\\dfrac{1}{4} \\cdot 2 \\cdot \\left( \\dfrac{3}{2} \\right)^3} = \\underline{\\dfrac{3 \\sqrt{3}}{4}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u66f2\u7dda \\(y = \\dfrac{1}{2} ( x^2 +1 )\\) \u4e0a\u306e\u70b9 P \u306b\u304a\u3051\u308b\u63a5\u7dda\u306f \\(x\\) \u8ef8\u3068\u4ea4\u308f\u308b\u3068\u3057, \u305d\u306e\u4ea4\u70b9\u3092 Q \u3068\u304a\u304f. \u7dda\u5206 PQ \u306e\u9577\u3055\u3092 \\(L\\) \u3068\u3059\u308b\u3068\u304d, \\(L\\) \u304c\u53d6\u308a … \u7d9a\u304d\u3092\u8aad\u3080