\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b.\r\n\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C\\) \u3092 \\(y = ax^3 -2x\\) \u3067\u5b9a\u3081\u308b.\r\n\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3068 \\(C\\) \u306e\u5171\u6709\u70b9\u306e\u500b\u6570\u304c \\(6\\) \u500b\u3067\u3042\u308b\u3088\u3046\u306a \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(C\\) \u3068\u5186\u306e\u5f0f\u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx^2 +( ax^3 -2x )^2 & = 1 \\\\\r\n\\text{\u2234} \\quad a^2 x^6 -4a x^4 +5x^2 -1 & = 0\r\n\\end{align}\\]\r\n\\(t = x^2\\) \u3068\u304a\u3051\u3070 \\(t \\geqq 0\\) \u3067\u3042\u308a\r\n\\[\r\na^2 t^3 -4a t^2 +5t -1 = 0 \\quad ... [1]\r\n\\]\r\n\u5de6\u8fba\u3092 \\(f(t)\\) \u3068\u304a\u3051\u3070, \\(f(0) = -1 \\neq 0\\) \u306a\u306e\u3067, [1] \u304c \\(t \\gt 0\\) \u306b\u7570\u306a\u308b \\(3\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf'(t) & = 3a^2 t^2 -8at +5 \\\\\r\n& = ( 3at -5 ) ( at -1 )\r\n\\end{align}\\]\r\n\\(f'(t) = 0\\) \u3092\u3068\u304f\u3068, \\(t = \\dfrac{1}{a} , \\dfrac{5}{3a}\\) .
\r\n\\(t \\gt 0\\) \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|cccccc} a & (0) & \\cdots & \\dfrac{1}{a} & \\cdots & \\dfrac{5}{3a} & \\cdots \\\\ \\hline f'(t) & & + & 0 & - & 0 & + \\\\ \\hline f(t) & (-1) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\nf \\left( \\dfrac{1}{a} \\right) \\gt 0 \\ \\text{\u304b\u3064} \\ f \\left( \\dfrac{5}{3a} \\right) \\lt 0\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{1}{a} \\right) & = \\dfrac{1}{a} -\\dfrac{4}{a} +\\dfrac{5}{a} -1 \\gt 0 \\\\\r\n& \\text{\u2234} \\quad a \\lt 2 \\ , \\\\\r\nf \\left( \\dfrac{5}{3a} \\right) & = \\dfrac{125}{27a} -\\dfrac{100}{9a} +\\dfrac{25}{3a} -1 \\lt 0 \\\\\r\n& \\text{\u2234} \\quad a \\gt \\dfrac{50}{27}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(a\\) \u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{\\dfrac{50}{27} \\lt a \\lt 2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u66f2\u7dda \\(C\\) \u3092 \\(y = ax^3 -2x\\) \u3067\u5b9a\u3081\u308b. \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u3068 \\(C\\) \u306e\u5171\u6709\u70b9\u306e\u500b\u6570\u304c \\(6\\) \u500b\u3067\u3042\u308b\u3088\u3046\u306a \\(a\\ … \u7d9a\u304d\u3092\u8aad\u3080