{"id":2765,"date":"2025-08-17T08:41:19","date_gmt":"2025-08-16T23:41:19","guid":{"rendered":"https:\/\/math-friend.com\/?p=2765"},"modified":"2025-08-18T09:43:02","modified_gmt":"2025-08-18T00:43:02","slug":"%e3%80%90%e5%90%8d%e5%8f%a4%e5%b1%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%a4%a7%e5%b0%8f%e6%af%94%e8%bc%83%e3%81%a8%e8%a7%a3%e3%81%a8%e4%bf%82%e6%95%b0%e3%81%ae","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2765","title":{"rendered":"\u3010\u540d\u53e4\u5c4b\u5927\u5b66\u5165\u8a66\u3011\u5bfe\u6570\u306e\u5927\u5c0f\u6bd4\u8f03\u3068\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2(2021)"},"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

4\u3064\u306e\u5b9f\u6570\\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\), \\(\\delta\\)\u3092, \\(\\displaystyle\\alpha=\\log_2{3}\\), \\(\\displaystyle\\beta=\\log_3{5}\\), \\(\\displaystyle\\gamma=\\log_5{2}\\), \\(\\displaystyle \\delta=\\frac{3}{2}\\)\u3068\u304a\u304f.
(1) \\(\\alpha\\beta\\gamma=1\\)\u3092\u793a\u305b.
(2) \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\), \\(\\delta\\)\u3092\u5c0f\u3055\u3044\u9806\u306b\u4e26\u3079\u3088.
(3) \\(p=\\alpha+\\beta+\\gamma\\), \\(\\displaystyle q=\\frac{1}{\\alpha}+\\frac{1}{\\beta}+\\frac{1}{\\gamma}\\)\u3068\u3057, \\(f(x)=x^3+px^2+qx+1\\)\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\displaystyle f\\left(-\\frac{1}{2}\\right)\\), \\(\\displaystyle f\\left(-1\\right)\\), \\(\\displaystyle f\\left(-\\frac{3}{2}\\right)\\)\u306e\u6b63\u8ca0\u3092\u5224\u5b9a\u305b\u3088.
(2021 \u540d\u53e4\u5c4b\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

(1) \\(\\beta\\), \\(\\gamma\\)\u306e\u5bfe\u6570\u306e\u5e95\u3092\\(2\\)\u306b\u5909\u63db\u3059\u308b\u3053\u3068\u3067, \\(\\alpha\\beta\\gamma\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u308b.$$
\\begin{align}
\\alpha\\beta\\gamma&=\\log_2{3}\\log_3{5}\\log_5{2}\\\\[1.5ex]
&=\\log_2{3}\\cdot\\frac{\\log_2{5}}{\\log_2{3}}\\cdot\\frac{\\log_2{2}}{\\log_2{5}}\\\\[1.5ex]
&=\\cancel{\\log_2{3}}\\cdot\\frac{\\cancel{\\log_2{5}}}{\\cancel{\\log_2{3}}}\\cdot\\frac{\\log_2{2}}{\\cancel{\\log_2{5}}}\\\\[1.5ex]
&=\\log_2{2}\\\\[1.5ex]
&=1
\\end{align}
$$
\u3088\u3063\u3066\u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) \u307e\u305a, \\(\\alpha>1\\), \\(\\beta>1\\), \\(\\gamma<1\\) \u3092\u793a\u3059. \u5404\u5bfe\u6570\u306e\u5e95\\(2\\), \\(3\\), \\(5\\)\u306f\u3044\u305a\u308c\u3082\\(1\\)\u3088\u308a\u5927\u304d\u3044\u306e\u3067, \u5bfe\u6570\u306e\u5927\u5c0f\u95a2\u4fc2\u3068, \u305d\u306e\u771f\u6570\u306e\u5927\u5c0f\u95a2\u4fc2\u304c\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066,
$$
\\begin{align}
\\alpha&=\\log_2{3}>\\log_2{2}=1\\\\[1.5ex]
\\beta&=\\log_3{5}>\\log_3{3}=1\\\\[1.5ex]
\\gamma&=\\log_5{2}<\\log_5{5}=1\\\\[1.5ex]
\\end{align}
$$\u3068\u306a\u308a, \u793a\u3055\u308c\u305f. \\(\\displaystyle\\delta=\\frac{3}{2}>1\\)\u306f\u660e\u3089\u304b\u306a\u306e\u3067, \u552f\u4e00\\(1\\)\u3088\u308a\u5c0f\u3055\u3044\\(\\gamma\\)\u304c\u6700\u5c0f\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\u6b21\u306b, \\(1\\)\u3088\u308a\u5927\u304d\u3044\\(\\alpha\\), \\(\\beta\\), \\(\\delta\\)\u306b\u95a2\u3057\u3066, \\(\\alpha\\)\u3068\\(\\delta\\), \\(\\beta\\)\u3068\\(\\delta\\)\u306e\u5927\u304d\u3055\u3092\u305d\u308c\u305e\u308c\u6bd4\u8f03\u3059\u308b.

\u30fb\\(\\alpha\\)\u3068\\(\\delta\\)\u306e\u6bd4\u8f03$$
\\begin{align}
2^{2\\alpha}&=\\left(2^\\alpha\\right)^2=3^2=9\\\\[1.5ex]
2^{2\\delta}&=2^{2\\cdot \\frac{3}{2}}=2^3=8
\\end{align}
$$\u3088\u308a, \\(2^{2\\alpha}>2^{2\\delta}\\)\u3067\u3042\u308a, \u5e95\u306e\\(2\\)\u306f\\(1\\)\u3088\u308a\u5927\u304d\u3044\u304b\u3089, \\(2\\alpha>2\\delta\\), \u3064\u307e\u308a, \\(\\alpha>\\delta\\)\u304c\u308f\u304b\u308b.

\u30fb\\(\\beta\\)\u3068\\(\\delta\\)\u306e\u6bd4\u8f03$$
\\begin{align}
3^{2\\beta}&=\\left(3^\\beta\\right)^2=5^2=25\\\\[1.5ex]
3^{2\\delta}&=3^{2\\cdot \\frac{3}{2}}=3^3=27
\\end{align}
$$\u3088\u308a, \\(3^{2\\beta}<3^{2\\delta}\\)\u3067\u3042\u308a, \u5e95\u306e\\(3\\)\u306f\\(1\\)\u3088\u308a\u5927\u304d\u3044\u304b\u3089, \\(2\\beta<2\\delta\\), \u3064\u307e\u308a, \\(\\beta<\\delta\\)\u304c\u308f\u304b\u308b.

\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068,$$
\\gamma<1<\\beta<\\delta<\\alpha
$$\u3068\u306a\u308b.<\/p>\n\n\n\n

(3) \\(\\alpha\\beta\\gamma=1\\)\u3088\u308a, \\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\)\u306f\u3044\u305a\u308c\u3082\\(0\\)\u3067\u306f\u306a\u3044\u306e\u3067, \u3053\u306e\u4e21\u8fba\u3092\\(\\alpha\\), \\(\\beta\\), \\(\\gamma\\)\u3067\u305d\u308c\u305e\u308c\u5272\u308b\u3053\u3068\u3067,$$
\\begin{align}
\\frac{1}{\\alpha}&=\\beta\\gamma\\\\[1.5ex]
\\frac{1}{\\beta}&=\\gamma\\alpha\\\\[1.5ex]
\\frac{1}{\\gamma}&=\\alpha\\beta
\\end{align}
$$\u3068\u306a\u308b. \u3053\u308c\u304b\u3089,$$
\\begin{align}
q&=\\frac{1}{\\alpha}+\\frac{1}{\\beta}+\\frac{1}{\\gamma}\\\\[1.5ex]
&=\\beta\\gamma+\\gamma\\alpha+\\alpha\\beta
\\end{align}
$$\u3068\u306a\u308a, \\(f(x)\\)\u306e\u5b9a\u6570\u9805\\(1\\)\u306f\\(\\alpha\\beta\\gamma\\)\u3068\u3082\u66f8\u3051\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089,
$$
f(x)=(x+\\alpha)(x+\\beta)(x+\\gamma)
$$\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u306e\u5f62\u304b\u3089, \\(\\displaystyle f\\left(-\\frac{1}{2}\\right)\\), \\(\\displaystyle f\\left(-1\\right)\\), \\(\\displaystyle f\\left(-\\frac{3}{2}\\right)\\)\u306e\u6b63\u8ca0\u3092\u5224\u5b9a\u3057\u3066\u3044\u304f.

\u30fb\\(\\displaystyle f\\left(-\\frac{1}{2}\\right)\\)\u306b\u3064\u3044\u3066
$$
f\\left(-\\frac{1}{2}\\right)=\\left(\\alpha-\\frac{1}{2}\\right)\\left(\\beta-\\frac{1}{2}\\right)\\left(\\gamma-\\frac{1}{2}\\right)
$$\u3067\u3042\u308a, \\(\\alpha>1\\), \\(\\beta>1\\)\u3088\u308a, \\(\\displaystyle \\alpha-\\frac{1}{2}\\), \\(\\displaystyle \\beta-\\frac{1}{2}\\)\u306f\u3044\u305a\u308c\u3082\u6b63\u3067\u3042\u308b.

\u3053\u3053\u3067, $$
5^\\gamma=2=\\sqrt{4}<\\sqrt{5}=5^\\frac{1}{2}
$$\u3067\u3042\u308a, \u5e95\\(5\\)\u306f\\(1\\)\u3088\u308a\u5927\u304d\u3044\u306e\u3067, \u3053\u308c\u304b\u3089, \\( \\displaystyle \\gamma<\\frac{1}{2}\\)\u304c\u308f\u304b\u308a, \\(\\displaystyle \\gamma-\\frac{1}{2}\\)\u306f\u8ca0\u306b\u306a\u308b.

\u4ee5\u4e0a\u304b\u3089, \\( \\displaystyle f\\left(-\\frac{1}{2}\\right)\\)\u306f\u300c(\u6b63)\u00d7(\u6b63)\u00d7(\u8ca0)\u300d\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067, \\( \\displaystyle f\\left(-\\frac{1}{2}\\right)<0\\)\u304c\u308f\u304b\u3063\u305f.

\u30fb\\(f\\left(-1\\right)\\)\u306b\u3064\u3044\u3066
$$
f\\left(-1\\right)=\\left(\\alpha-1\\right)\\left(\\beta-1\\right)\\left(\\gamma-1\\right)
$$\u3068\u8868\u305b\u308b. \\(\\alpha>1\\), \\(\\beta>1\\)\u3088\u308a, \\(\\displaystyle \\alpha-1\\), \\(\\displaystyle \\beta-1\\)\u306f\u3044\u305a\u308c\u3082\u6b63\u3067\u3042\u308a, \\(\\gamma<1\\)\u3088\u308a, \\(\\displaystyle \\gamma-1\\)\u306f\u8ca0\u3067\u3042\u308b.

\u4ee5\u4e0a\u304b\u3089, \\( \\displaystyle f\\left(-1\\right)\\)\u306f\u300c(\u6b63)\u00d7(\u6b63)\u00d7(\u8ca0)\u300d\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067, \\( \\displaystyle f\\left(-1\\right)<0\\)\u304c\u308f\u304b\u3063\u305f.

\u30fb\\(\\displaystyle f\\left(-\\frac{3}{2}\\right)\\)\u306b\u3064\u3044\u3066
\\(\\displaystyle\\delta=\\frac{3}{2}\\)\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \\(\\displaystyle f\\left(-\\frac{3}{2}\\right)\\)\u306f,
$$
\\begin{align}
f\\left(-\\frac{3}{2}\\right)&=\\left(\\alpha-\\frac{3}{2}\\right)\\left(\\beta-\\frac{3}{2}\\right)\\left(\\gamma-\\frac{3}{2}\\right)\\\\[1.5ex]
&=(\\alpha-\\delta)(\\beta-\\delta)(\\gamma-\\delta)
\\end{align}
$$\u3068\u8868\u305b\u308b. \u3053\u3053\u3067, \\(\\alpha>\\delta\\) \u3088\u308a, \\(\\displaystyle \\alpha-\\delta\\)\u306f\u6b63\u3067\u3042\u308a, \\(\\beta<\\delta\\), \\(\\gamma<\\delta\\)\u3088\u308a, \\(\\displaystyle \\beta-\\delta\\), \\(\\displaystyle \\gamma-\\delta\\)\u306f\u3044\u305a\u308c\u3082\u8ca0\u3067\u3042\u308b.

\u4ee5\u4e0a\u304b\u3089, \\( \\displaystyle f\\left(-\\frac{3}{2}\\right)\\)\u306f\u300c(\u6b63)\u00d7(\u8ca0)\u00d7(\u8ca0)\u300d\u3068\u306a\u3063\u3066\u3044\u308b\u306e\u3067, \\( \\displaystyle f\\left(-\\frac{3}{2}\\right)>0\\)\u304c\u308f\u304b\u3063\u305f.<\/p>\n<\/div><\/div>\n\n\n\n

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