{"id":2904,"date":"2025-08-27T02:41:26","date_gmt":"2025-08-26T17:41:26","guid":{"rendered":"https:\/\/math-friend.com\/?p=2904"},"modified":"2025-08-27T02:42:08","modified_gmt":"2025-08-26T17:42:08","slug":"%e3%80%90%e7%ad%91%e6%b3%a2%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e9%96%a2%e6%95%b0%e3%81%8c%e6%a5%b5%e5%80%a4%e3%82%92%e3%81%a8%e3%82%89%e3%81%aa%e3%81%84%e3%82%88%e3%81%86%e3%81%ab%e3%81%99","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2904","title":{"rendered":"\u3010\u7b51\u6ce2\u5927\u5b66\u5165\u8a66\u3011\u95a2\u6570\u304c\u6975\u5024\u3092\u3068\u3089\u306a\u3044\u3088\u3046\u306b\u3059\u308b\u554f\u984c(2024)"},"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

\\(a\\), \\(b\\)\u3092\u5b9f\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u95a2\u6570\\(f(x)\\)\u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.$$
f(x)=(1-2x^2)\\cos{2x}+2x\\sin{2x}+a\\cos^2{x}+b\\int_0^x{t\\sin{2t}}\\,dt
$$
(1) \\(a=8\\pi^2\\), \\(b=-4\\pi\\)\u306e\u3068\u304d, \\(\\displaystyle 0<x<\\frac{3}{2}\\pi\\)\u306e\u7bc4\u56f2\u3067\\(f(x)\\)\u304c\u6975\u5024\u3092\u3068\u308b\u3088\u3046\u306a\\(x\\)\u306e\u5024\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.
(2) \u6761\u4ef6:\u300c\\(f(x)\\)\u304c\\(0<x<\\frac{3}{2}\\pi\\)\u306e\u7bc4\u56f2\u3067\u6975\u5024\u3092\u3068\u3089\u306a\u3044\u300d\u3092\u6e80\u305f\u3059\\(a\\), \\(b\\)\u3092\u6c42\u3081\u3088.
(2024 \u7b51\u6ce2\u5927\u5b66 \u7406\u7cfb [5])<\/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

(1) \\(f(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068,$$
\\begin{align}
f^\\prime(x)&=-4x\\cos{2x}-2(1-2x^2)\\sin{2x}+2\\sin{2x}+4x\\cos{2x}-2a\\cos{x}\\sin{x}+bx\\sin{2x}\\\\[1.5ex]
&=(4x^2+bx-a)\\sin{2x}
\\end{align}
$$\u3068\u306a\u308a, \\(a=8\\pi^2\\), \\(b=-4\\pi\\)\u3092\u4ee3\u5165\u3059\u308b\u3068,$$
f^\\prime(x)=(4x^2-4\\pi x-8\\pi^2)\\sin{2x}=4(x+\\pi)(x-2\\pi)\\sin{2x}
$$\u3068\u306a\u308b. \\(\\displaystyle 0<x<\\frac{3}{2}\\pi\\)\u306b\u304a\u3044\u3066, \\(f^\\prime(x)=0\\)\u3068\u3059\u308b\u3068, \\(\\displaystyle x=\\frac{\\pi}{2}, \\pi\\)\u3068\u306a\u308b. \u3053\u308c\u304b\u3089\u5897\u6e1b\u8868\u3092\u66f8\u304f\u3068,$$
\\begin{array}{|c|c|c|c|c|c|c|c|}
\\hline
x & 0 & \\cdots & \\frac{\\pi}{2} & \\cdots & \\pi & \\cdots & \\frac{3}{2}\\pi \\\\[1.5ex]
\\hline
f^\\prime(x) & & – & 0 & + & 0 & – & \\\\[1.5ex]
\\hline
f(x) & & \\searrow & \u6975\u5c0f &\\nearrow& \u6975\u5927 &\\searrow &\\\\[1.5ex]
\\hline
\\end{array}$$
\u3068\u306a\u308a, \\(\\displaystyle x=\\frac{\\pi}{2}, \\pi\\)\u3067\u6975\u5024\u3092\u3068\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

(2) (1)\u304b\u3089\\(f^\\prime(x)=(4x^2+bx-a)\\sin{2x}\\)\u3067\u3042\u308a, \\(\\displaystyle 0<x<\\frac{3}{2}\\pi\\)\u306e\u7bc4\u56f2\u306b\u304a\u3044\u3066, \\(\\sin{2x}\\)\u306f\\(x=\\frac{\\pi}{2}, \\pi\\)\u306e\u524d\u5f8c\u3067\u7b26\u53f7\u3092\u5909\u3048\u308b.

\u3053\u3053\u3067, \\(f(x)\\)\u304c\\(x=c\\)\u3067\u6975\u5024\u3092\u3068\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \\(f^\\prime(x)\\)\u304c\\(x=c\\)\u306e\u524d\u5f8c\u3067\u7b26\u53f7\u3092\u5909\u3048\u308b\u3053\u3068\u3067\u3042\u308b. \u3088\u3063\u3066, \\(f(x)\\)\u304c\u6975\u5024\u3092\u3068\u3089\u306a\u3044\u3088\u3046\u306b\u3059\u308b\u305f\u3081\u306b\u306f, \\(4x^2+bx-a\\)\u3082\u307e\u305f\\(\\displaystyle x=\\frac{\\pi}{2}, \\pi\\)\u306e\u524d\u5f8c\u3067\u7b26\u53f7\u3092\u5909\u3048\u308b\u5fc5\u8981\u304c\u3042\u308a, \\(4x^2+bx-a\\), \\(\\sin{2x}\\)\u304c\u5171\u306b\\(\\displaystyle x=\\frac{\\pi}{2}, \\pi\\)\u306e\u524d\u5f8c\u3067\u7b26\u53f7\u3092\u5909\u3048\u308b\u3053\u3068\u3067, \\(f^\\prime(x)\\)\u306e\u7b26\u53f7\u5909\u5316\u304c\u6253\u3061\u6d88\u3055\u308c\u308b.

\u3088\u3063\u3066, \\(f(x)\\)\u306f$$
\\begin{align}
f(x)&=4x^2+bx-a\\\\[1.5ex]
&=4\\left(x-\\frac{\\pi}{2}\\right)\\left(x-\\pi\\right)\\\\[1.5ex]
&=4x^2-6\\pi x+2\\pi^2
\\end{align}
$$\u306e\u5f62\u3067\u8868\u3055\u308c\u308b\u5fc5\u8981\u304c\u3042\u308a, \\(a=-2\\pi^2\\), \\(b=-6\\pi\\)\u3068\u306a\u308b.

\u9006\u306b, \\(a=-2\\pi^2\\), \\(b=-6\\pi\\)\u306e\u3068\u304d, \u5897\u6e1b\u8868\u306f,
$$
\\begin{array}{|c|c|c|c|c|c|c|c|}
\\hline
x & 0 & \\cdots & \\frac{\\pi}{2} & \\cdots & \\pi & \\cdots & \\frac{3}{2}\\pi \\\\[1.5ex]
\\hline
f^\\prime(x) & & + & 0 & + & 0 & + & \\\\[1.5ex]
\\hline
f(x) & & \\nearrow & &\\nearrow& &\\nearrow &\\\\[1.5ex]
\\hline
\\end{array}$$
\u3068\u306a\u308a, \\(f^\\prime(x)\\)\u306f\\(\\displaystyle 0<x<\\frac{3}{2}\\pi\\)\u306e\u7bc4\u56f2\u3067\u7b26\u53f7\u3092\u5909\u3048\u306a\u3044\u3053\u3068\u3082\u308f\u304b\u308a, \\(f(x)\\)\u306f \\(0<x<\\frac{3}{2}\\pi\\)\u306e\u7bc4\u56f2\u3067\u6975\u5024\u3092\u3068\u3089\u306a\u3044. \u4ee5\u4e0a\u304b\u3089, \\(a=-2\\pi^2\\), \\(b=-6\\pi\\)\u3068\u6c42\u307e\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

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