{"id":1947,"date":"2025-07-27T19:59:07","date_gmt":"2025-07-27T10:59:07","guid":{"rendered":"https:\/\/math-friend.com\/?p=1947"},"modified":"2025-08-01T10:03:46","modified_gmt":"2025-08-01T01:03:46","slug":"%e3%80%90%e4%b8%80%e6%a9%8b%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e5%85%b1%e6%9c%89%e7%82%b9%e3%81%a7%e6%8e%a5%e7%b7%9a%e3%81%8c%e7%9b%b4%e4%ba%a4%e3%81%99%e3%82%8b2%e3%81%a4%e3%81%ae2%e6%ac%a1","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1947","title":{"rendered":"\u3010\u4e00\u6a4b\u5927\u5b66\u5165\u8a66\u3011\u5171\u6709\u70b9\u3067\u63a5\u7dda\u304c\u76f4\u4ea4\u3059\u308b2\u3064\u306e2\u6b21\u95a2\u6570\u306b\u56f2\u307e\u308c\u308b\u9762\u7a4d(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\u3057\u3066, 2\u3064\u306e\u66f2\u7dda\\(C_1: y=x^2\\)\u3068\\(C_2:y=-x^2+ax+b\\)\u306f\u5171\u6709\u70b9\u3092\u6301\u3061, \u305d\u306e\u70b9\u306b\u304a\u3051\u308b\u305d\u308c\u305e\u308c\u306e\u63a5\u7dda\u306f\u76f4\u4ea4\u3057\u3066\u3044\u308b. \u3053\u306e\u3088\u3046\u306a\u6761\u4ef6\u3092\u6e80\u305f\u3059\\(C_1\\), \\(C_2\\)\u306b\u95a2\u3057\u3066, \\(C_1\\), \\(C_2\\)\u3067\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.
(2024 \u4e00\u6a4b\u5927\u5b66 [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

\\(f(x)=x^2\\), \\(g(x)=-x^2+ax+b\\)\u3068\u304a\u304f. \u6761\u4ef6\u3092\u6e80\u305f\u3059\\(C_1\\), \\(C_2\\)\u306e\u5171\u6709\u70b9\u306e\\(x\\)\u5ea7\u6a19\u3092\\(t\\)\u3068\u3059\u308b\u3068,
$$
\\begin{align}
f(t)=g(t) &\\iff t^2=-t^2+at+b \\\\[1.5ex]
&\\iff 2t^2-at-b=0\\,\\,\\,\u30fb\u30fb\u30fb\u2460
\\end{align}
$$\u304c\u6210\u308a\u7acb\u3064. \u307e\u305f, \u305d\u306e\u5171\u6709\u70b9\u306b\u304a\u3051\u308b\u5404\u66f2\u7dda\u306e\u63a5\u7dda\u306e\u50be\u304d\u306f\\(f^\\prime(t)=2t\\), \\(g^\\prime(t)=-2t+a\\)\u3067\u3042\u308a, \u3053\u308c\u304c\u76f4\u4ea4\u3059\u308b\u304b\u3089,
$$
\\begin{align}
f^\\prime(t)\\cdot g^\\prime(t)=-1 & \\iff 2t\\cdot (-2t+a)=-1\\\\[1.5ex]
&\\iff 4t^2-2at-1=0\\,\\,\\,\u30fb\u30fb\u30fb\u2461
\\end{align}
$$\u3068\u306a\u308b. \\(a\\),\\(b\\)\u306b\u5bfe\u3057\u3066\u2460, \u2461\u3092\u540c\u6642\u306b\u6e80\u305f\u3059\\(t\\)\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.

\\(2\\times \u2460-\u2461\\)\u3088\u308a,
$$
-2b+1=0\\iff b=\\frac{1}{2}
$$\u3068\u306a\u308a, \u2460, \u2461\u3092\u540c\u6642\u306b\u6e80\u305f\u3059\\(t\\)\u304c\u5b58\u5728\u3059\u308b\u305f\u3081\u306b\u306f\\( \\displaystyle b=\\frac{1}{2}\\)\u304c\u5fc5\u8981\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\\(\\displaystyle b=\\frac{1}{2}\\)\u3067\u3042\u308c\u3070, \u2460\u3092\\(t\\)\u306e2\u6b21\u65b9\u7a0b\u5f0f\u3068\u898b\u3066\u5224\u5225\u5f0f\\(D\\)\u3092\u8a08\u7b97\u3059\u308b\u3068,
$$
D=(-a)^2-4\\cdot 2\\cdot \\left(-\\frac{1}{2}\\right)=a^2+4>0
$$\u3068\u306a\u308a, \u2460\u306f\\(a\\)\u306e\u5024\u306b\u4f9d\u3089\u305a2\u3064\u306e\u5b9f\u6570\u89e3\u3092\u6301\u3064\u3053\u3068\u304c\u308f\u304b\u308b. \u305d\u3057\u3066\u307e\u305f, \\(\\displaystyle b=\\frac{1}{2}\\)\u3067\u3042\u308c\u3070, \u2460\u3068\u2461\u304c\u540c\u4e00\u306e\u65b9\u7a0b\u5f0f\u3068\u306a\u308b\u304b\u3089, \u2460\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f\u540c\u6642\u306b\u2461\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u3068\u306a\u308b.

\u4ee5\u4e0a\u304b\u3089, \\(\\displaystyle b\\neq\\frac{1}{2}\\)\u3067\u3042\u308c\u3070, \\(C_1\\), \\(C_2\\)\u306f\u5171\u6709\u70b9\u3092\u6301\u3063\u305f\u3068\u3057\u3066\u3082, \u305d\u306e\u5171\u6709\u70b9\u3067\u63a5\u7dda\u304c\u76f4\u4ea4\u3059\u308b\u3053\u3068\u304c\u306a\u3044\u3053\u3068, \u305d\u3057\u3066, \\(\\displaystyle b=\\frac{1}{2}\\)\u3067\u3042\u308c\u3070, \\(a\\)\u306e\u5024\u306b\u4f9d\u3089\u305a, \\(C_1\\), \\(C_2\\)\u306f2\u3064\u306e\u5171\u6709\u70b9\u3092\u6301\u3061, \u3055\u3089\u306b\u305d\u306e\u5171\u6709\u70b9\u306b\u304a\u3051\u308b2\u3064\u306e\u63a5\u7dda\u306f\u76f4\u4ea4\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f.

\u4ee5\u964d, \\(\\displaystyle b=\\frac{1}{2}\\)\u3068\u3059\u308b.

\u2460\u3092\u89e3\u3044\u3066\\(C_1\\), \\(C_2\\)\u306e\u5171\u6709\u70b9\u306e\\(x\\)\u5ea7\u6a19\u3092\\(a\\)\u3092\u7528\u3044\u3066\u8868\u3059\u3068,
$$
\\begin{align}
2t^2-at-\\frac{1}{2}=0 &\\iff 4t^2-2at-1=0 \\\\[1.5ex]
& \\iff t=\\frac{a\\pm\\sqrt{a^2+4}}{4}
\\end{align}
$$\u3068\u306a\u308b. \u3053\u306e\u89e3\u3092\u305d\u308c\u305e\u308c\\(\\displaystyle \\alpha=\\frac{a-\\sqrt{a^2+4}}{4}\\), \\(\\displaystyle \\beta=\\frac{a +\\sqrt{a^2+4}}{4}\\)\u3068\u3059\u308b\u3068, \u3053\u308c\u306f\\(g(x)=f(x)\\)\u306e\u89e3\u3067\u3082\u3042\u308b\u304b\u3089,
$$
g(x)-f(x)=-x^2+ax+b-x^2=-2(x-\\alpha)(x-\\beta)
$$\u3068\u8868\u305b\u308b.

\\(C_1\\), \\(C_2\\)\u306e\u66f2\u7dda\u306e\u4f4d\u7f6e\u95a2\u4fc2\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a,

\u3053\u308c\u304b\u3089, \\(C_1\\), \\(C_2\\)\u3067\u56f2\u307e\u308c\u308b\u9818\u57df\u306e\u9762\u7a4d\\(S(a)\\)\u306f,
$$
\\begin{align}
S(a)&=\\int_\\alpha^\\beta\\left\\{g(x)-f(x)\\right\\}\\,dx\\\\[1.5ex]
&=-2\\int_\\alpha^\\beta(x-\\alpha)(x-\\beta)\\,dx\\\\[1.5ex]
&=-2\\cdot\\left(-\\frac{1}{6}\\right)(\\beta-\\alpha)^3\\\\[1.5ex]
&=\\frac{1}{3}\\left(\\frac{\\sqrt{a^2+4}}{2}\\right)^3\\\\[1.5ex]
&=\\frac{1}{24}(a^2+4)^{\\frac{3}{2}}
\\end{align}
$$\u3068\u306a\u308a, \u3053\u308c\u304c\u6700\u5c0f\u3068\u306a\u308b\u306e\u306f\\(a=0\\)\u306e\u3068\u304d\u3067\u3042\u308b.\u305d\u306e\u6700\u5c0f\u5024\\(S(0)\\)\u306f,
$$
S(0)=\\frac{1}{24}\\cdot 4^{\\frac{3}{2}}=\\frac{1}{3}
$$\u3068\u6c42\u307e\u308b.
<\/p>\n\n\n\n

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