{"id":1922,"date":"2021-10-09T09:42:38","date_gmt":"2021-10-09T00:42:38","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1922"},"modified":"2021-10-09T09:42:38","modified_gmt":"2021-10-09T00:42:38","slug":"ykr201604","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201604\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2016\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b.\r\n\\(2\\) \u3064\u306e\u66f2\u7dda \\(C _ 1 : \\ y = x \\log x\\) \u3068 \\(C _ 2 : \\ y = ax^2\\) \u306e\u4e21\u65b9\u306b\u63a5\u3059\u308b\u76f4\u7dda\u306e\u672c\u6570\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(\\displaystyle\\lim _ {x \\rightarrow \\infty} \\dfrac{( \\log x )^2}{x} = 0\\) \u306f\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

\\(C_1\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = 1 \\cdot \\log x +x \\cdot \\dfrac{1}{x} = 1 +\\log x\r\n\\]\r\n\\(x\\) \u5ea7\u6a19\u304c \\(t \\ ( t \\gt 0 )\\) \u306e\u70b9\u306b\u304a\u3051\u308b \\(C_1\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = ( 1 +\\log t ) ( x-t ) +t \\log t \\\\\r\n& = ( 1 +\\log t ) x -t \\quad ... [1]\r\n\\end{align}\\]\r\n\\(C_2\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = 2ax\r\n\\]\r\n\\(x\\) \u5ea7\u6a19\u304c \\(s\\) \u306e\u70b9\u306b\u304a\u3051\u308b \\(C_2\\) \u306e\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = 2as ( x-s ) +as^2 \\\\\r\n& = 2as x -as^2 \\quad ... [2]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] [2] \u304c\u4e00\u81f4\u3059\u308b \\(( s , t )\\) \u306e\u7d44\u306e\u500b\u6570\u304c, \u6c42\u3081\u308b\u672c\u6570\u3067\u3042\u308b.
\r\n[1] [2] \u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\r\n\\left\\{ \\begin{array}{ll} 1 +\\log t = 2as & \\quad ... [3] \\\\ t = as^2 & \\quad ... [4] \\end{array} \\right.\r\n\\]\r\n[4] \u3088\u308a, \\(s = \\pm \\sqrt{\\dfrac{t}{a}}\\) \u3067, [3] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n1 +\\log t & = \\pm 2 \\sqrt{at} \\\\\r\n\\dfrac{1 +\\log t}{\\sqrt{t}} & = \\pm 2 \\sqrt{a}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(u = \\sqrt{t}\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\dfrac{1 +2 \\log u}{u} = \\pm 2 \\sqrt{a} \\quad ... [5]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [5] \u3092\u307f\u305f\u3059 \\(u\\) \u306e\u500b\u6570\u304c, \u6c42\u3081\u308b\u672c\u6570\u3067\u3042\u308b.
\r\n[5] \u306e\u5de6\u8fba\u3092 \\(f(u)\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nf'(u) & = \\dfrac{\\dfrac{2}{u} \\cdot u -( 1 +2 \\log u ) \\cdot 1}{u^2} \\\\\r\n& = \\dfrac{1 -2 \\log u}{u^2}\r\n\\end{align}\\]\r\n\\(f'(u) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\n\\log u & = \\dfrac{1}{2} \\\\\r\n\\text{\u2234} \\quad u & = \\sqrt{e}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\nf \\left( \\sqrt{e} \\right) = \\dfrac{2}{\\sqrt{e}}\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nf(u) & = \\dfrac{1}{u} +\\dfrac{\\log u^2}{u} \\\\\r\n& \\rightarrow 0+0 = 0 \\quad ( \\ u \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\ , \\\\\r\nf(u) & = \\dfrac{1 +2 \\log u}{u} \\rightarrow -\\infty \\quad ( \\ u \\rightarrow 0 \\ \\text{\u306e\u3068\u304d} \\ )\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(f(u)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} u & ( 0 ) & \\cdots & \\sqrt{e} & \\cdots & ( \\infty ) \\\\ \\hline f'(u) & & + & 0 & - & \\\\ \\hline f(u) & ( -\\infty ) & \\nearrow & \\dfrac{2}{\\sqrt{e}} & \\searrow & ( 0 ) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u672c\u6570\u306f<\/p>\r\n