\\(p , q\\) \u3092\u5b9a\u6570\u3068\u3057, \\(0 \\lt p \\lt 1\\) \u3068\u3059\u308b.\r\n\\[\\begin{align}\r\n\\text{\u66f2\u7dda} \\ C_1 \\ & : \\ y = p x^{\\frac{1}{p}} \\quad ( x \\gt 0 ) \\quad \\text{\u3068, } \\\\\r\n\\text{\u66f2\u7dda} \\ C_2 \\ & : \\ y = \\log x +q \\quad ( x \\gt 0 )\r\n\\end{align}\\]\r\n\u304c, \u3042\u308b \\(1\\) \u70b9 \\(( a , b )\\) \u306b\u304a\u3044\u3066\u540c\u3058\u76f4\u7dda\u306b\u63a5\u3059\u308b\u3068\u3059\u308b.\r\n\u66f2\u7dda \\(C_1\\) , \u76f4\u7dda \\(x = a\\) , \u76f4\u7dda \\(x = e^{-q}\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S_1\\) \u3068\u3059\u308b.\r\n\u307e\u305f, \u66f2\u7dda \\(C_2\\) , \u76f4\u7dda \\(x = a\\) \u304a\u3088\u3073 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u306e\u9762\u7a4d\u3092 \\(S_2\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(q\\) \u3092 \\(p\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(S_1 , S_2\\) \u3092 \\(p\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(\\dfrac{S_2}{S_1} \\geqq \\dfrac{3}{4}\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b. \u305f\u3060\u3057, \\(2.5 \\lt e \\lt 3\\) \u3092\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(0 \\lt p \\lt 1\\) \u3088\u308a, \\(\\dfrac{1}{p} \\gt 1\\) . (2)<\/strong><\/p>\r\n \\(a = 1\\) \u3088\u308a, \\(b = p\\) . (3)<\/strong><\/p>\r\n \\(S_1 \\gt 0\\) \u306a\u306e\u3067, \u793a\u3057\u305f\u3044\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\n4 S_2 -3 S_1 \\geqq 0 \\quad ... [\\text{A}]\r\n\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3092\u793a\u305b\u3070\u3088\u3044.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(C_1\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = p \\cdot \\dfrac{1}{p} x^{\\frac{1}{p} -1} = x^{\\frac{1}{p} -1}\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 \\(( a , b )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = a^{\\frac{1}{p} -1} ( x-a ) +p a^{\\frac{1}{p}} \\\\\r\n& = a^{\\frac{1}{p} -1} x -( 1-p ) a^{\\frac{1}{p}} \\quad ... [1]\r\n\\end{align}\\]\r\n\\(C_2\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = \\dfrac{1}{x}\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 \\(( a , b )\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\dfrac{1}{a} ( x-a ) +\\log a +q \\\\\r\n& = \\dfrac{1}{a} x +\\log a +q -1 \\quad ... [2]\r\n\\end{align}\\]\r\n[1] [2] \u304c\u4e00\u81f4\u3059\u308b\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{ll} a^{\\frac{1}{p} -1} = \\dfrac{1}{a} & ... [3] \\\\ -( 1-p ) a^{\\frac{1}{p}} = \\log a +q -1 & ... [4] \\end{array} \\right.\r\n\\]\r\n[3] \u3088\u308a\r\n\\[\\begin{align}\r\na^{\\frac{1}{p}} & = 1 \\\\\r\n\\text{\u2234} \\quad a & = 1\r\n\\end{align}\\]\r\n[4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\np-1 & = q-1 \\\\\r\n\\text{\u2234} \\quad q & = \\underline{p}\r\n\\end{align}\\]\r\n
\r\n\\(0 \\lt p \\lt 1\\) \u3088\u308a, \\(\\dfrac{1}{e} \\lt e^{-p} \\lt 1\\) .
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS_1 & = \\displaystyle\\int _ {e^{-p}}^{1} p x^{\\frac{1}{p}} \\, dx \\\\\r\n& = p \\left[ \\dfrac{1}{\\frac{1}{p} +1} x^{\\frac{1}{p} +1} \\right] _ {e^{-p}}^{1} \\\\\r\n& = \\underline{\\dfrac{p^2}{p+1} \\left( 1 -e^{-p-1} \\right)}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nS_2 & = \\displaystyle\\int _ {e^{-p}}^{1} ( \\log x +p ) \\, dx \\\\\r\n& = \\left[ x ( \\log x -1) +px \\right] _ {e^{-p}}^{1} \\\\\r\n& = -1 +p +e^{-p} ( p+1 ) -p e^{-p} \\\\\r\n& = \\underline{e^{-p} +p -1}\r\n\\end{align}\\]\r\n
\r\n\\[\\begin{align}\r\n4 S_2 -3 S_1 & = 4 \\left( e^{-p} +p -1 \\right) -\\dfrac{3 p^2}{p+1} \\left( 1 -e^{-p-1} \\right) \\\\\r\n& = \\dfrac{4 ( p+1 ) \\left( e^{-p} +p -1 \\right) -3 p^2 ( 1 -e^{-p-1} )}{p+1} \\\\\r\n& = \\dfrac{\\left( \\dfrac{3}{e} p^2 +4p +4 \\right) e^{-p} +p^2 -4}{p+1} \\\\\r\n& \\gt \\dfrac{( p+2 )^2 e^{-p} +( p+2 ) ( p-2 )}{p+1} \\quad \\left( \\ \\text{\u2235} \\ \\dfrac{3}{e} \\gt 1 \\ \\right) \\\\\r\n& = \\dfrac{p+2}{p+1} \\underline{\\left\\{ (p+2) e^{-p} +p-2 \\right\\}} _ {[5]}\r\n\\end{align}\\]\r\n[5] \u3092 \\(f(p)\\) \u3068\u304a\u304f\u3068, \\(f(p) \\gt 0\\) \u3092\u793a\u305b\u3070\u3088\u3044. \r\n\\[\\begin{align}\r\nf'(p) & = e^{-p} -( p+2 ) e^{-p} +1 \\\\\r\n& = -( p+1 ) e^{-p} +1 \\ , \\\\\r\nf''(p) & = -e^{-p} +( p+1 ) e^{-p} =p e^{-p} \\gt 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f'(p)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057\r\n\\[\r\nf'(p) \\gt f'(0) = -1 +1 = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(p)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057\r\n\\[\r\nf(p) \\gt f(0) = 2 -2 = 0\r\n\\]\r\n\u3088\u3063\u3066, [A] \u304c\u793a\u3055\u308c\u3066\u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(p , q\\) \u3092\u5b9a\u6570\u3068\u3057, \\(0 \\lt p \\lt 1\\) \u3068\u3059\u308b. \\[\\begin{align} \\text{\u66f2\u7dda} \\ C_1 \\ & : \\ y = p x^{\\frac{1}{p}} \\quad ( … \u7d9a\u304d\u3092\u8aad\u3080