{"id":3131,"date":"2025-09-13T06:00:00","date_gmt":"2025-09-12T21:00:00","guid":{"rendered":"https:\/\/math-friend.com\/?p=3131"},"modified":"2025-09-13T10:45:05","modified_gmt":"2025-09-13T01:45:05","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e5%af%be%e6%95%b0%e3%81%ae%e5%85%a5%e3%81%a3%e3%81%9f%e9%96%a2%e6%95%b0%e3%81%8c%e3%81%82%e3%82%8b%e7%af%84%e5%9b%b2%e3%81%a7","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3131","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66\u5165\u8a66\u3011\u5b9a\u7a4d\u5206\u306e\u6761\u4ef6\u304b\u3089\u6700\u5927\u5024\u3092\u6c42\u3081\u308b\u554f\u984c(2008)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(0\\leq \\alpha \\leq \\beta\\)\u3092\u307f\u305f\u3059\u5b9f\u6570\\(\\alpha\\), \\(\\beta\\)\u3068, \\(2\\)\u6b21\u5f0f\\(f(x)=x^2-(\\alpha+\\beta)x+\\alpha\\beta\\)\u306b\u3064\u3044\u3066,$$
\\int_{-1}^1f(x)\\,dx=1
$$\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u5b9a\u7a4d\u5206$$
S=\\int_0^\\alpha f(x)\\,dx
$$\u3092\\(\\alpha\\)\u306e\u5f0f\u3067\u8868\u3057, \\(S\\)\u304c\u3068\u308a\u3046\u308b\u5024\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.
(2000 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [1])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

\\(\\displaystyle \\int_{-1}^1f(x)\\,dx=1\\)\u3088\u308a,
$$
\\begin{align}
\\int_{-1}^1f(x)\\,dx&=\\int_{-1}^1\\left\\{x^2-(\\alpha+\\beta)x+\\alpha\\beta\\right\\}\\,dx\\\\[1.5ex]
&=\\left[\\frac{x^3}{3}-(\\alpha+\\beta)\\frac{x^2}{2}+\\alpha\\beta x\\right]_{-1}^1\\\\[1.5ex]
&=\\frac{2}{3}+2\\alpha\\beta=1
\\end{align}
$$\u3068\u306a\u308a, \\(\\displaystyle \\alpha\\beta=\\frac{1}{6}\\)\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

(\u88dc\u8db3) \u5076\u95a2\u6570\\(f(x)\\)\u306b\u5bfe\u3057\u3066,$$
\\int_{-a}^af(x)\\,dx=2\\int_0^af(x)\\,dx
$$\u304c, \u5947\u95a2\u6570\\(g(x)\\)\u306b\u5bfe\u3057\u3066, $$
\\int_{-a}^ag(x)\\,dx=0
$$\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066,$$
\\int_{-1}^1\\left\\{x^2-(\\alpha+\\beta)x+\\alpha\\beta\\right\\}\\,dx=2\\int_0^1(x^2-\\alpha\\beta)\\,dx
$$\u3068\u3057\u3066\u8a08\u7b97\u3057\u3066\u3082\u826f\u3044\u3067\u3059.<\/p>\n\n\n\n

\u3053\u3053\u3067, \\(\\alpha=0\\)\u3068\u3059\u308b\u3068, \\(\\alpha\\beta=0\\)\u3068\u306a\u3063\u3066\u3057\u307e\u3046\u305f\u3081, \\(\\alpha > 0\\)\u304c\u308f\u304b\u308b. \u3053\u308c\u304b\u3089, \\(\\displaystyle \\beta=\\frac{1}{6\\alpha}\\)\u3067\u3042\u308b.

\\(\\alpha\\leq \\beta\\)\u3088\u308a, \\(\\displaystyle \\alpha\\leq \\frac{1}{6\\alpha}\\)\u3067\u3042\u308a, \\(\\alpha>0\\)\u306b\u6ce8\u610f\u3057\u3066, \\( \\displaystyle \\alpha^2\\leq\\frac{1}{6}\\)\u3068\u306a\u308b. \u3088\u3063\u3066\\(\\alpha\\)\u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f\\( \\displaystyle 0<\\alpha\\leq\\frac{1}{\\sqrt{6}}\\)\u3067\u3042\u308b.

\\(S\\)\u3092\u8a08\u7b97\u3059\u308b\u3068,$$
\\begin{align}
S&=\\int_0^\\alpha f(x)\\,dx\\\\[1.5ex]
&=\\int_0^\\alpha \\left\\{ x^2-(\\alpha+\\beta)x+\\alpha\\beta \\right\\}\\,dx\\\\[1.5ex]
&=\\left[\\frac{x^3}{3}-(\\alpha+\\beta)\\frac{x^2}{2}+\\alpha\\beta x\\right]_0^\\alpha\\\\[1.5ex]
&=\\frac{\\alpha^3}{3}-(\\alpha+\\beta)\\frac{\\alpha^2}{2}+\\alpha^2\\beta\\\\[1.5ex]
&=-\\frac{\\alpha^3}{6}+\\frac{\\alpha^2\\beta}{2}
\\end{align}
$$\u3067\u3042\u308a, \\(\\displaystyle \\beta=\\frac{1}{6\\alpha}\\)\u3088\u308a,$$
S=-\\frac{\\alpha^3}{6}+\\frac{\\alpha}{12}
$$\u3068\u3057\u3066, \\(S\\)\u3092\\(\\alpha\\)\u306e\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n\n\n\n

(\u88dc\u8db3) \u6570\u2161\u307e\u3067\u306e\u7bc4\u56f2\u3067\u306f\u7fd2\u3044\u307e\u305b\u3093\u304c, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u90e8\u5206\u7a4d\u5206\u3092\u4f7f\u3063\u3066\u3082\u826f\u3044\u3067\u3059.$$
\\begin{align}
S&=\\int_0^\\alpha(x-\\alpha)(x-\\beta)\\,dx\\\\[1.5ex]
&=\\left[\\frac{1}{2}(x-\\alpha)^2(x-\\beta)\\right]_0^\\alpha-\\frac{1}{2}\\int_0^\\alpha (x-\\alpha)^2\\,dx\\\\[1.5ex]
&=\\frac{\\alpha^2\\beta}{2}-\\frac{1}{6}\\left[(x-\\alpha)^3\\right]_0^\\alpha\\\\[1.5ex]
&=\\frac{\\alpha^2\\beta}{2}-\\frac{\\alpha^3}{6}
\\end{align}
$$<\/p>\n\n\n\n

\\(S(\\alpha)=-\\frac{\\alpha^3}{6}+\\frac{\\alpha}{12}\\)\u3068\u3057\u3066, \\(S\\)\u3092\\(\\alpha\\)\u306e\u95a2\u6570\u3068\u307f\u306a\u3057, \\(S(\\alpha)\\)\u3092\\(\\alpha\\)\u3067\u5fae\u5206\u3059\u308b\u3068,$$
\\begin{align}
S^\\prime(\\alpha)&=-\\frac{\\alpha^2}{2}+\\frac{1}{12}\\\\[1.5ex]
&=-\\frac{1}{2}\\left(\\alpha^2-\\frac{1}{6}\\right)
\\end{align}
$$\u3068\u306a\u308a, \\(S^\\prime(\\alpha)=0\\)\u3068\u3059\u308b\u3068, \\(\\displaystyle \\alpha=\\pm\\frac{1}{\\sqrt{6}}\\)\u3067\u3042\u308b.

\\( \\displaystyle 0<\\alpha\\leq\\frac{1}{\\sqrt{6}}\\)\u306e\u7bc4\u56f2\u3067, \\(S(\\alpha)\\)\u306e\u5897\u6e1b\u8868\u306f,$$
\\begin{array}{|c|c|c|c|}
\\hline
\\alpha & 0 & \\cdots & \\frac{1}{\\sqrt{6}} \\\\[1.5ex]
\\hline
S^\\prime(\\alpha) & \\cancel{\\phantom{\\Large 0}} & + & 0 \\\\[1.5ex]
\\hline
S(\\alpha) & \\cancel{\\phantom{\\Large 0}} & \\nearrow & \\\\[1.5ex]
\\hline
\\end{array}$$\u3068\u306a\u308a, \\(S(\\alpha)\\)\u306f\\(\\displaystyle \\alpha=\\frac{1}{\\sqrt{6}}\\)\u306e\u3068\u304d, \u6700\u5927\u5024$$
\\begin{align}
S\\left(\\frac{1}{\\sqrt{6}}\\right)&=-\\frac{1}{6^2\\sqrt{6}}+\\frac{1}{12\\sqrt{6}}\\\\[1.5ex]
&=-\\frac{\\sqrt{6}}{6^3}+\\frac{\\sqrt{6}}{12\\cdot 6}\\\\[1.5ex]
&=\\frac{\\sqrt{6}}{6^3}(-1+3)\\\\[1.5ex]
&=\\frac{\\sqrt{6}}{108}
\\end{align}
$$\u3092\u3068\u308b\u3053\u3068\u304c\u308f\u304b\u308b. <\/p>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n