{"id":3120,"date":"2025-09-11T10:29:54","date_gmt":"2025-09-11T01:29:54","guid":{"rendered":"https:\/\/math-friend.com\/?p=3120"},"modified":"2025-09-11T10:35:02","modified_gmt":"2025-09-11T01:35:02","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e9%83%bd%e7%ab%8b%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","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3120","title":{"rendered":"\u3010\u6771\u4eac\u90fd\u7acb\u5927\u5b66\u5165\u8a66\u3011\u5bfe\u6570\u306e\u5165\u3063\u305f\u95a2\u6570\u304c\u3042\u308b\u7bc4\u56f2\u3067\u8ca0\u306b\u306a\u308b\u6761\u4ef6(2020)"},"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\\)\u3092\u5b9f\u6570\u3068\u3057\u3066, \u95a2\u6570\\(f(x)\\)\u3092
\\begin{equation} f(x)=\\frac{1}{4}\\left(\\log_{\\frac{1}{2}}{x^2}\\right)^2-\\log_{\\frac{1}{2}}{\\left(8x^{1-a}\\right)}+a-8 \\end{equation}
\u3068\u3059\u308b. \\(\\displaystyle t=\\log_{\\frac{1}{2}}{x}\\)\u3068\u304a\u304f\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(f(x)\\)\u3092\\(t\\)\u3092\u7528\u3044\u3066\u8868\u305b.
(2) \\(1\\leq x\\leq 4\\)\u306e\u3068\u304d, \\(t\\)\u306e\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.
(3) \u6761\u4ef6\u300c\\(1\\leq x\\leq 4\\)\u306e\u3068\u304d, \\(f(x)<0\\)\u3067\u3042\u308b\u300d\u3092\u6e80\u305f\u3059\\(a\\)\u306e\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.
(2020 \u6771\u4eac\u90fd\u7acb\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

(1) \u4e00\u822c\u306b\\(a>1\\), \\(x<0\\)\u306e\u3068\u304d, \\(x^{1-a}\\)\u306f\u5b9a\u7fa9\u3055\u308c\u306a\u3044\u306e\u3067, \u771f\u6570\u6761\u4ef6\u3068\u5408\u308f\u305b\u3066\\(f(x)\\)\u306e\u5b9a\u7fa9\u57df\\(x>0\\)\u3067\u3042\u308b. \\(x>0\\)\u306a\u306e\u3067, \\(\\displaystyle \\log_{\\frac{1}{2}}{x^2}=2\\log_{\\frac{1}{2}}{x}\\), \\(\\displaystyle \\log_{\\frac{1}{2}}{x^{1-a}}=(1-a)\\log_{\\frac{1}{2}}{x}\\)\u3067\u3042\u308b\u304b\u3089, $$
\\begin{align}
f(x)&=\\frac{1}{4}\\left(2\\log_{\\frac{1}{2}}{x}\\right)^2-\\log_{\\frac{1}{2}}{8}-(1-a)\\log_{\\frac{1}{2}}{x}+a-8\\\\[1.5ex]
&=t^2-(-3)-(1-a)t+a-8\\\\[1.5ex]
&=t^2-(1-a)t+a-5
\\end{align}
$$\u3068\u306a\u308b.<\/p>\n\n\n\n

(2) \\(t=\\log_{\\frac{1}{2}}{x}\\)\u306e\u30b0\u30e9\u30d5\u304b\u3089, \\(1\\leq x\\leq 4\\)\u306e\u3068\u304d, \\(-2\\leq t\\leq 0\\)\u3067\u3042\u308b.
<\/p>\n\n\n\n

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

(3) \\(g(t)=t^2-(1-a)t+a-5\\)\u3068\u304a\u304f\u3068, (1), (2)\u3088\u308a, \u6761\u4ef6\u306f\u300c\\(-2\\leq t\\leq 0\\)\u306e\u3068\u304d, \\(g(t)<0\\)\u3067\u3042\u308b\u300d\u3068\u540c\u5024\u3067\u3042\u308b. \\(y=g(t)\\)\u306f\u4e0b\u306b\u51f8\u306e\u653e\u7269\u7dda\u306e\u305f\u3081, \u3053\u308c\u306f, \u300c\\(g(-2)<0\\) \u304b\u3064 \\(g(0)<0\\)\u300d\u3068\u540c\u5024\u3067\u3042\u308b. $$
\\begin{align}
g(-2)&=4+2(1-a)+a-5<0\\\\[1.5ex]
&\\iff a>1\\\\[1.5ex]
g(0)&=a-5<0\\\\[1.5ex]
&\\iff a<5
\\end{align} \u3088\u308a, \u6c42\u3081\u308b\\(a\\)\u306e\u7bc4\u56f2\u306f\\(1<a<5\\)\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