\\(xyz\\) \u7a7a\u9593\u3067, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(\\sqrt{6}\\) \u306e\u7403\u9762 \\(S\\) \u3068 \\(3\\) \u70b9 \\(( 4 , 0 , 0 )\\) , \\(( 0 , 4 , 0 )\\) ,\r\n\\(( 0 , 0 , 4 )\\) \u3092\u901a\u308b\u5e73\u9762 \\(\\alpha\\) \u304c\u5171\u6709\u70b9\u3092\u6301\u3064\u3053\u3068\u3092\u793a\u3057,\r\n\u70b9 \\(( x , y , z )\\) \u304c\u305d\u306e\u5171\u6709\u70b9\u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, \u7a4d \\(xyz\\) \u304c\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(3\\) \u70b9\u3092\u901a\u308b\u5e73\u9762\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\\begin{align}\r\n\\dfrac{x}{4} +\\dfrac{y}{4} +\\dfrac{z}{4} & = 1 \\\\\r\n\\text{\u2234} \\quad x+y+z & = 4 \\quad ... [1]\r\n\\end{align}\\]\r\n\u539f\u70b9\u304b\u3089\u3053\u306e\u5e73\u9762\u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u306e\u8db3\u3092 H \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{OH} = \\dfrac{\\large| 0+0+0-4 \\large|}{\\sqrt{1^2+1^2+1^2}} = \\dfrac{4}{\\sqrt{3}}\r\n\\]\r\n\\(\\dfrac{4^2}{3} \\lt 6\\) \u3088\u308a \\(\\text{OH} \\lt \\sqrt{6}\\) \u306a\u306e\u3067, \\(\\alpha\\) \u3068 \\(S\\) \u306f\u5171\u6709\u70b9\u3092\u3082\u3064.
\r\n\u5171\u6709\u70b9\u306f [1] \u3068 \\(x^2+y^2+z^2 = 6\\) \u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nx^2+y^2+z^2 & = ( x+y+z )^2 +2( xy+yz+zx ) \\\\\r\n& = 4^2 +2( xy+yz+zx ) = 6 \\\\\r\n\\text{\u2234} & \\quad xy+yz+zx = 5\r\n\\end{align}\\]\r\n\\(k = xyz\\) \u3068\u304a\u304f\u3068, \\(x , y , z\\) \u306f\u65b9\u7a0b\u5f0f \\(t^3 -4t^2 +5t -k = 0\\) , \u3059\u306a\u308f\u3061\r\n\\[\r\nt^3 -4t^2 +5t = k\r\n\\]\r\n\u306e \\(3\\) \u3064\u306e\u5b9f\u89e3\u3067\u3042\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066, \u3053\u306e\u65b9\u7a0b\u5f0f\u304c \\(3\\) \u3064\u306e\u5b9f\u89e3\u3092\u3082\u3064, \u3064\u307e\u308a, \u30b0\u30e9\u30d5 \\(u=f(t) = t^3 -4t^2 +5t \\quad ... [2]\\) \u3068\u76f4\u7dda \\(u=k\\) \u304c\u5171\u6709\u70b9\u3092 \\(3\\) \u3064\uff08\u63a5\u70b9\u306f \\(2\\) \u3064\u3068\u6570\u3048\u308b\uff09\u3082\u3064\u305f\u3081\u306e \\(k\\) \u306e\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n[2] \u3088\u308a\r\n\\[\r\nf'(t) = 3t^2 -8t +5 = ( t-1 )( 3t-5 )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066 \\(f(t)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & \\cdots & 1 & \\cdots & \\frac{5}{3} & \\cdots \\\\ \\hline f'(t) & + & 0 & - & 0 & + \\\\ \\hline f(t) & \\nearrow & 2 & \\searrow & \\frac{50}{27} & \\nearrow \\\\ \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u7bc4\u56f2\u306f, \\(f\\left( \\dfrac{5}{3} \\right) \\leqq k \\leqq f(1)\\) , \u3059\u306a\u308f\u3061\r\n\\[\r\n\\underline{\\dfrac{50}{27} \\leqq xyz \\leqq 2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u3067, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(\\sqrt{6}\\) \u306e\u7403\u9762 \\(S\\) \u3068 \\(3\\) \u70b9 \\(( 4 , 0 , 0 )\\) , \\(( 0 , 4 , 0 )\\) , \\(( 0 , 0 … \u7d9a\u304d\u3092\u8aad\u3080