{"id":1375,"date":"2025-07-18T20:01:38","date_gmt":"2025-07-18T11:01:38","guid":{"rendered":"https:\/\/math-friend.com\/?p=1375"},"modified":"2025-08-04T23:50:59","modified_gmt":"2025-08-04T14:50:59","slug":"%e3%80%90%e5%8c%97%e6%b5%b7%e9%81%93%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%913%e6%ac%a1%e9%96%a2%e6%95%b0%e3%81%ae%e6%a5%b5%e5%80%a4%e3%81%a8%e6%8e%a5%e7%b7%9a2025","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=1375","title":{"rendered":"\u3010\u5317\u6d77\u9053\u5927\u5b66\u5165\u8a66\u30113\u6b21\u95a2\u6570\u306e\u6975\u5024\u3068\u63a5\u7dda(2025)"},"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

3\u6b21\u95a2\u6570\\(f(x)=x^3-6x^2-15x+30\\)\u306b\u3064\u3044\u3066, \u66f2\u7dda\\(C:y=f(x)\\)\u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(f(x)\\)\u306e\u6975\u5024\u3092\u6c42\u3081\u3088.
(2) \\(C\\)\u4e0a\u306e\u70b9\\((-3, -6)\\)\u3092\u901a\u308a, \\(C\\)\u306b\u63a5\u3059\u308b\u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u5168\u3066\u6c42\u3081\u3088.
(2025 \u5317\u6d77\u9053\u5927\u5b66 \u6587\u7cfb[1])<\/span><\/p>\n\n\n\n

\u3053\u306e\u554f\u984c\u306f\u6559\u79d1\u66f8\u306e\u4f8b\u984c\u30ec\u30d9\u30eb\u306e\u8d85\u57fa\u672c\u554f\u984c\u306a\u306e\u3067\u78ba\u5b9f\u306b\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046.<\/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)=x^3-6x^2-15x+30\\)\u3092\u5fae\u5206\u3059\u308b\u3068,
$$
f^\\prime(x)=3x^2-12x-15=3(x^2-4x-5)=3(x-5)(x+1)
$$\u3068\u306a\u308a, \\(f^\\prime(x)=0\\)\u3068\u304a\u304f\u3068, \\(x=-1, 5\\)\u3068\u306a\u308b. \u3053\u3053\u3067, \\(f(-1)=38\\), \\(f(5)=-70\\)\u306b\u6ce8\u610f\u3057\u3066, \u5897\u6e1b\u8868\u3092\u66f8\u304f\u3068,
$$
\\begin{array}{|c|c|c|c|c|c|}
\\hline
x & \\cdots & -1 &\\cdots & 5 & \\cdots \\\\[1.5ex]
\\hline
f'(x) & + & 0 & – & 0 & + \\\\[1.5ex]
\\hline
f(x) & \\nearrow & 38 & \\searrow & -70 & \\nearrow \\\\[1.5ex]
\\hline
\\end{array}
$$\u3068\u306a\u308b.

\u3088\u3063\u3066\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a, \\(f(x)\\)\u306f\\(x=-1\\)\u3067\u6975\u5927\u5024\\(28\\)\u3092, \\(x=5\\)\u3067\u6975\u5c0f\u5024\\(-20\\)\u3092\u3068\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

(2) \u4efb\u610f\u306e\u5b9f\u6570\\(a\\)\u306b\u5bfe\u3057\u3066, \\(C\\)\u306e\\(x=a\\)\u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3068, \u3053\u308c\u306f\u50be\u304d\u304c\\(f^\\prime(a)\\)\u3067\\((a, f(a))\\)\u3092\u901a\u308b\u3053\u3068\u304b\u3089,
$$
\\begin{align}
y&=f^\\prime(a)(x-a)+f(a)\\\\[1.5ex]
&=(3a^2-12a-15)(x-a)+a^3-6a^2-15a+30\\\\[1.5ex]
&=(3a^2-12a-15)x-2a^3+6a^2+30
\\end{align}
$$\u3068\u306a\u308b. \u3053\u308c\u304c\\((-3, -6)\\)\u3092\u901a\u308b\u304b\u3089,
$$
\\begin{align}
&-6=(3a^2-12a-15)\\cdot(-3)-2a^3+6a^2+30\\\\[1.5ex]
&\\iff 2a^3+3a^2-36a-81=0
\\end{align}
$$\u3068\u3057\u3066\\(a\\)\u306e3\u6b21\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u308b. \\(a=-3\\)\u306f\u3053\u306e\u65b9\u7a0b\u5f0f\u306e1\u3064\u306e\u89e3\u3067\u3042\u308a, \u5de6\u8fba\u306f\\((a+3)\\)\u3092\u56e0\u5b50\u306b\u6301\u3064\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u304b\u3089\u65b9\u7a0b\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068,
$$
\\begin{align}
&2a^3+3a^2-36a-81=0\\\\[1.5ex]
&\\iff (a+3)(2a^2-3a-27)=0\\\\[1.5ex]
&\\iff (a+3)^2(2a-9)=0
\\end{align}
$$\u3068\u306a\u308a, \u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f,
$$
\\begin{align}
a&=-3\\,\\,(\u91cd\u89e3)\\\\[1.5ex]
a&=\\frac{9}{2}
\\end{align}$$\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u304b\u3089\u6c42\u3081\u308b\u63a5\u7dda\u306f, \\(a=-3\\)\u306e\u3068\u304d,
$$
\\begin{align}
y&=\\left(3\\cdot(-3)^2-12\\cdot(-3)-15\\right)x-2\\cdot(-3)^3+6\\cdot(-3)^2+30\\\\[1.5ex]
&=48x+138
\\end{align}
$$\u3067\u3042\u308a, \\( \\displaystyle a=\\frac{9}{2}\\)\u306e\u3068\u304d,
$$
\\begin{align}
y&=\\left(3\\cdot\\left(\\frac{9}{2}\\right)^2-12\\cdot\\left(\\frac{9}{2}\\right)-15\\right)x-2\\cdot\\left(\\frac{9}{2}\\right)^3+6\\cdot\\left(\\frac{9}{2}\\right)^2+30\\\\[1.5ex]
&=-\\frac{33}{4}x-\\frac{123}{4}
\\end{align}$$
\u3068\u306a\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u5b9f\u969b\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u308b\u3068, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a, \u6c42\u3081\u305f2\u672c\u306e\u76f4\u7dda\u304c\\((-3, -6)\\)\u3092\u901a\u308a\\(C\\)\u306b\u63a5\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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