(1)<\/strong>\u3000\\(f'(a) = 0\\) , \\(f''(b) = 0\\) \u3092\u6e80\u305f\u3059 \\(a , b\\) \u3092\u6c42\u3081, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u63cf\u3051. \u305f\u3060\u3057, \\(\\displaystyle\\lim _ {x \\rightarrow \\infty} \\sqrt{x} e^{-x} = 0\\) \u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u306b\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(k \\geqq 0\\) \u306e\u3068\u304d \\(V(k) = \\displaystyle\\int _ {0}^{k} x e^{-2x} \\, dx\\) \u3092 \\(k\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000(1)<\/strong> \u3067\u6c42\u3081\u305f \\(a , b\\) \u306b\u5bfe\u3057\u3066\u66f2\u7dda \\(y = f(x)\\) \u3068 \\(x\\) \u8ef8\u304a\u3088\u3073 \\(2\\) \u76f4\u7dda \\(x=a\\) , \\(x=b\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{\\sqrt{x}} \\cdot e^{-x} -2 \\sqrt{x} e^{-x} \\\\\r\n& = \\dfrac{( 1 -2x ) e^{-x}}{\\sqrt{x}} \\ .\r\n\\end{align}\\]\r\n\\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = a = \\underline{\\dfrac{1}{2}} \\ .\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nf''(x) & = -\\dfrac{1}{2 x^{\\frac{3}{2}}} \\cdot e^{-x} -\\dfrac{2 ( 1 -2x ) e^{-x}}{\\sqrt{x}} \\\\\r\n& = \\dfrac{( 4x^2 -4x +1 ) e^{-x}}{2 x^{\\frac{3}{2}}} \\\\\r\n& = \\dfrac{\\left( x -\\frac{1 -\\sqrt{2}}{2} \\right) \\left( x -\\frac{1 +\\sqrt{2}}{2} \\right) e^{-x}}{2 x^{\\frac{3}{2}}} \\ .\r\n\\end{align}\\]\r\n\\(x \\geqq 0\\) \u306b\u304a\u3044\u3066, \\(f'(x) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nx = b = \\underline{\\dfrac{1 +\\sqrt{2}}{2}} \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n& f(0) = 0 \\ , \\quad \\displaystyle\\lim _ {x \\rightarrow \\infty} f(x) = 0 \\ , \\\\\r\n& \\displaystyle\\lim _ {x \\rightarrow +0} f'(x) = \\displaystyle\\lim _ {x \\rightarrow +0} \\left( \\dfrac{1}{\\sqrt{x}} -2 \\sqrt{x} \\right) e^{-x} = \\infty\r\n\\end{align}\\]\r\n \u3057\u305f\u304c\u3063\u3066, \\(f(x)\\) \u306e\u5897\u6e1b, \u51f9\u51f8\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} x & 0 & \\cdots & \\dfrac{1}{2} & \\cdots & \\dfrac{1 +\\sqrt{2}}{2} & \\cdots & ( \\infty ) \\\\ \\hline f'(x) & ( \\infty ) & + & 0 & - & & - & \\\\ \\hline f''(x) & & - & & - & 0 & + & \\\\ \\hline f(x) & 0 & \\nearrow ( \\cap ) & & \\searrow ( \\cap ) & & \\searrow ( \\cup ) & (0) \\end{array}\r\n\\]\r\n\\[\r\nf \\left( \\dfrac{1}{2} \\right) = \\sqrt{\\dfrac{2}{e}} \\ , \\ f \\left( \\dfrac{1 +\\sqrt{2}}{2} \\right) = \\sqrt{2 +2 \\sqrt{2}} e^{\\frac{1 +\\sqrt{2}}{2}}\r\n\\]\r\n\u306a\u306e\u3067, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u306f\u4e0b\u56f3\u306e\u901a\u308a.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(G(x) = \\displaystyle\\int x e^{-2x} \\, dx\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nG(x) & = -\\dfrac{1}{2} x e^{-2x} +\\dfrac{1}{2} \\displaystyle\\int e^{-2x} \\, dx \\\\\r\n& = -\\dfrac{1}{2} x e^{-2x} -\\dfrac{1}{4} e^{-2x} \\\\\r\n& = -\\dfrac{2x +1}{4} e^{-2x} +C \\quad ( \\ C \\ \\text{\u306f\u7a4d\u5206\u5b9a\u6570} \\ ) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nV(k) & = G(k) -G(0) \\\\\r\n& = -\\dfrac{2k +1}{4} e^{-2k} +\\dfrac{1}{4} \\\\\r\n& = \\underline{\\dfrac{1}{4} \\left\\{ 1 -( 2k +1 ) e^{-2k} \\right\\}} \\ .\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u4f53\u7a4d \\(W\\) \u306f\r\n\\[\\begin{align}\r\nW & = 4 \\pi \\displaystyle\\int _ {0}^{b} x e^{-2x} \\, dx -4 \\pi \\displaystyle\\int _ {0}^{a} x e^{-2x} \\, dx \\\\\r\n& = 4 \\pi \\left\\{ V(b) -V(a) \\right\\} \\\\\r\n& = \\pi \\left\\{ 1 -( 2b+1 ) e^{-2b} -1 +( 2a+1 ) e^{-2a}\\right\\} \\\\\r\n& = \\underline{\\pi \\left\\{ 2 e^{-1} -( 2 +\\sqrt{2} ) e^{-1 -\\sqrt{2}} \\right\\}} \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) = 2 \\sqrt{x} e^{-x} \\ ( x \\geqq 0 )\\) \u306b\u3064\u3044\u3066\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(f'(a) = 0\\) , \\(f''(b) = 0\\) \u3092\u6e80\u305f\u3059 \\(a , b\\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
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