{"id":3364,"date":"2025-09-28T01:38:52","date_gmt":"2025-09-27T16:38:52","guid":{"rendered":"https:\/\/math-friend.com\/?p=3364"},"modified":"2025-09-28T01:38:56","modified_gmt":"2025-09-27T16:38:56","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6-%e6%96%87%e7%b3%bb1-2010%e3%80%91%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%a8%e4%b8%89%e8%a7%92%e5%bd%a2%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%81%ae%e6%9c%80-2","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3364","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66 \u6587\u7cfb1 (2010)\u3011\u4e09\u89d2\u95a2\u6570\u3068\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u6700\u5927\u5316"},"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

\\(2\\)\u6b21\u95a2\u6570\\(f(x)=x^2+ax+b\\)\u306b\u5bfe\u3057\u3066, $$
f(x+1)=c\\int_0^1(3x^2+4xt)f^\\prime(t)\\,dt
$$\u304c\\(x\\)\u306b\u3064\u3044\u3066\u306e\u6052\u7b49\u5f0f\u3068\u306a\u308b\u3088\u3046\u306a\u5b9a\u6570\\(a\\), \\(b\\), \\(c\\)\u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.
(2010 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [2])<\/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

\\(f^\\prime(x)=2x+a\\)\u3088\u308a, \\(f(x+1)=c\\int_0^1(3x^2+4xt)f^\\prime(t)\\,dt\\)\u306f,$$
\\begin{align}
(x+1)^2+a(x+1)+b=c\\int_0^1(3x^2+4xt)(2t+a)\\,dt
\\end{align}
$$\u3068\u306a\u308a, \u5de6\u8fba\u3092\u5c55\u958b, \u53f3\u8fba\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057, \u6574\u7406\u3059\u308b\u3068,$$
\\begin{align}
x^2+(a+2)x+a+b+c&=c\\int_0^1(6x^2t+3ax^2+8xt^2+4axt)\\,dt\\\\[1.5ex]
&=c\\left[3x^2t^2+3ax^2t+\\frac{8}{3}xt^3+2axt^2\\right]_0^1\\\\[1.5ex]
&=c\\left(3x^2+3ax^2+\\frac{8}{3}x+2ax\\right)\\\\[1.5ex]
&=3c(a+1)x^2+c\\left(\\frac{8}{3}+2a\\right)x
\\end{align}
$$\u3068\u306a\u308b. \u3053\u308c\u304c\u6052\u7b49\u5f0f\u306b\u306a\u308b\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \u4e21\u8fba\u3067\\(x^2\\), \\(x\\)\u306e\u4fc2\u6570, \u304a\u3088\u3073\u5b9a\u6570\u9805\u304c\u7b49\u3057\u3044\u3053\u3068\u3067\u3042\u308b\u304b\u3089, \u3053\u306e\u6761\u4ef6\u306f$$
\\left\\{
\\begin{align}
&1=3c(a+1)\\,\\,\\,\u30fb\u30fb\u30fb\u2460\\\\[1.5ex]
&a+2=c\\left(\\frac{8}{3}+2a\\right)\\,\\,\\,\u30fb\u30fb\u30fb\u2461\\\\[1.5ex]
&a+b+1=0\\,\\,\\,\u30fb\u30fb\u30fb\u2462\\end{align}\\right.
$$\u3068\u306a\u308b.

\u2460, \u2461\u3092\u6574\u7406\u3059\u308b\u3068,$$
\\begin{align}
&3c+3ac=1\\,\\,\\, \\cdots \u2460^\\prime\\\\[1.5ex]
&8c+6ac=3a+6\\,\\,\\,\\ \\cdots \u2461^\\prime
\\end{align}
$$\u3068\u306a\u308a, \\(ac\\)\u3092\u6d88\u53bb\u3059\u308b\u305f\u3081\u306b, \\(2\\times \u2460^\\prime – \\times \u2461^\\prime\\)\u3092\u8a08\u7b97\u3057\u3066,$$
\\begin{align}
-2c=-3a-4\\\\[1.5ex]
\\end{align}
$$\u3068\u306a\u308a, \u3053\u308c\u304b\u3089,$$
c=\\frac{3a}{2}+2
$$\u3068\u306a\u308b. \u3053\u308c\u3092\u2460\u306b\u4ee3\u5165\u3057\u3066,$$
\\begin{align}
&1=3\\left(\\frac{3a}{2}+2\\right)(a+1)\\\\[1.5ex]
\\iff &2=3(3a+4)(a+1)\\\\[1.5ex]
\\iff &9a^2+21a+10=0\\\\[1.5ex]
\\iff &(3a+2)(3a+5)=0\\\\[1.5ex]
\\iff &a=-\\frac{2}{3},\\,\\,-\\frac{5}{3}
\\end{align}
$$\u3068\u306a\u308b. \u3053\u308c\u304b\u3089, \\(\\displaystyle a=-\\frac{2}{3}\\)\u306e\u3068\u304d, \u2462\u304b\u3089\\(\\displaystyle b=-\\frac{1}{3}\\), \\(\\displaystyle c=\\frac{3a}{2}+2\\)\u304b\u3089, \\(c=1\\)\u3068\u306a\u308b. \u307e\u305f, \\(\\displaystyle a=-\\frac{5}{3}\\)\u306e\u3068\u304d, \u2462\u304b\u3089\\(\\displaystyle b=\\frac{2}{3}\\), \\(\\displaystyle c=\\frac{3a}{2}+2\\)\u304b\u3089, \\(\\displaystyle c=-\\frac{1}{2}\\)\u3068\u306a\u308b.

\u3088\u3063\u3066, \u6c42\u3081\u308b\\(a\\), \\(b\\), \\(c\\)\u306e\u7d44\u306f,$$
(a,b,c)=\\left(-\\frac{2}{3},-\\frac{1}{3},1\\right), \\left(-\\frac{5}{3},\\frac{2}{3},-\\frac{1}{2}\\right)
$$\u3068\u306a\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

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