\u95a2\u6570 \\(f(x) = e^{ax} \\ (a \\gt 0 )\\) \u3068\u6b21\u306e\u6761\u4ef6 (\u30a2)<\/strong> , (\u30a4)<\/strong> \u3092\u6e80\u305f\u3059\u95a2\u6570 \\(g(x)\\) \u304c\u3042\u308b.<\/p>\r\n (\u30a2)<\/strong>\u3000\\(y = g(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u534a\u5186\r\n\\[\r\n\\left\\{\\begin{array}{l} (x-p)^2 +(y-q)^2 = r^2 \\\\ y \\lt p \\end{array}\\right.\r\n\\]\r\n\u3067\u3042\u308b. \u305f\u3060\u3057, \\(p \\lt 0\\) , \\(q \\gt 0\\) , \\(r \\gt |p|\\) \u3068\u3059\u308b.<\/p><\/li>\r\n (\u30a4)<\/strong>\u3000\\(f(0) = g(0)\\) , \\(f'(0) = g'(0)\\) , \\(f''(0) = g''(0)\\)<\/p><\/li>\r\n<\/ol>\r\n \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n (1)<\/strong>\u3000\\(p , q , r\\) \u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(a\\) \u304c\u3059\u3079\u3066\u306e\u6b63\u306e\u5b9f\u6570\u3092\u52d5\u304f\u3068\u304d, \\(r\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6 (\u30a2)<\/strong> \u3088\u308a\r\n\\[\r\ng(x) = q -\\sqrt{r^2 -(x-p)^2}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\ng'(x) = \\dfrac{x-p}{\\sqrt{r^2 -(x-p)^2}}\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\ng''(x) & = \\dfrac{\\sqrt{r^2 -(x-p)^2} -(x-p) \\cdot \\frac{x-p}{\\sqrt{r^2 -(x-p)^2}}}{r^2 -(x-p)^2} \\\\\r\n& = \\dfrac{\\sqrt{r^2 -(x-p)^2} -(x-p) g'(x)}{r^2 -(x-p)^2}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\nf'(x) = a e^{ax} , \\ f''(x) = a^2 e^{ax}\r\n\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6 (\u30a4)<\/strong> \u3088\u308a\r\n\\[\r\n\\left\\{ \\begin{array}{ll} q-s = 1 & ... [1] \\\\ -\\dfrac{p}{s} = a & ... [2] \\\\ \\dfrac{s+pa}{s^2} = a^2 & ... [3] \\end{array} \\right.\r\n\\]\r\n\u305f\u3060\u3057, \u3053\u3053\u3067 \\(s = \\sqrt{r^2-p^2}\\) ... [4] \u3068\u304a\u3044\u305f. (2)<\/strong><\/p>\r\n \\(h(a) = r^2 = \\dfrac{\\left( 1 +a^2 \\right)^3}{a^4}\\) \u3068\u304a\u3044\u3066, \\(h(a)\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nh'(a) & = \\dfrac{3 \\left( 1 +a^2 \\right)^2 2a \\cdot a^4 -4a^3 \\left( 1 +a^2 \\right)^3}{a^8} \\\\\r\n& = \\dfrac{2 \\left( a^2 -2 \\right) \\left( 1 +a^2 \\right)^2}{a^5}\r\n\\end{align}\\]\r\n\\(h'(a) = 0\\) \u3092\u3068\u304f\u3068, \\(a \\gt 0\\) \u306a\u306e\u3067\r\n\\[\r\na = \\sqrt{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(h(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} a & (0) & \\cdots & \\sqrt{2} & \\cdots \\\\ \\hline h'(a) & & - & 0 & + \\\\ \\hline h(a) & & \\searrow & \\text{\u6700\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na = \\underline{\\sqrt{2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) = e^{ax} \\ (a \\gt 0 )\\) \u3068\u6b21\u306e\u6761\u4ef6 (\u30a2) , (\u30a4) \u3092\u6e80\u305f\u3059\u95a2\u6570 \\(g(x)\\) \u304c\u3042\u308b. (\u30a2)\u3000\\(y = g(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u534a\u5186 \\[ \\left\\{\\beg … \u7d9a\u304d\u3092\u8aad\u3080 \r\n
\r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n[2] \u3088\u308a, \\(p = -sa\\) \u306a\u306e\u3067, [3] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{gather}\r\n\\dfrac{1+a^2}{s} = a^2 \\\\\r\n\\text{\u2234} \\quad s = 1 +\\dfrac{1}{a^2}\r\n\\end{gather}\\]\r\n[1] [2] [4] \u3088\u308a\r\n\\[\\begin{align}\r\np & = -sa = \\underline{-a -\\dfrac{1}{a}} , \\\\\r\nq & = 1+s = \\underline{2 +\\dfrac{1}{a^2}} , \\\\\r\nr & = \\sqrt{p^2 +s^2} \\\\\r\n& = \\sqrt{\\dfrac{\\left( 1+a^2 \\right)^2}{a^2} +\\dfrac{\\left( 1+a^2 \\right)^2}{a^4}} \\\\\r\n& = \\underline{\\dfrac{\\left( 1+a^2 \\right)^{\\frac{3}{2}}}{a^2}}\r\n\\end{align}\\]\r\n