\\(a , b , c\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3057, \\(x\\) \u306e\u95a2\u6570 \\(y = x^3 +ax^2 +bx +c\\) \u3092\u8003\u3048\u308b.\r\n\u4ee5\u4e0b, \u5b9a\u6570\u306f\u3059\u3079\u3066\u5b9f\u6570\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u5b9a\u6570 \\(p , q\\) \u306b\u5bfe\u3057, \u6b21\u3092\u307f\u305f\u3059\u5b9a\u6570 \\(r\\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\nx \\geqq 1\\quad \\text{\u306a\u3089\u3070} \\quad \\left| px +q \\right| \\leqq rx\n\\]<\/li>\r\n (2)<\/strong>\u3000\u6052\u7b49\u5f0f \\(( \\alpha -\\beta )( \\alpha^2 +\\alpha \\beta +\\beta^2 ) = \\alpha^3 -\\beta^3\\) \u3092\u7528\u3044\u3066, \u6b21\u3092\u307f\u305f\u3059\u5b9a\u6570 \\(k , l\\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\nx \\geqq 1 \\quad \\text{\u306a\u3089\u3070} \\quad \\left| \\sqrt[3]{f(x)} -x -k \\right| \\leqq \\dfrac{l}{x}\n\\]<\/li>\r\n (3)<\/strong>\u3000\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(\\sqrt[3]{f(n)}\\) \u304c\u81ea\u7136\u6570\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u95a2\u6570 \\(f(x)\\) \u306f, \u81ea\u7136\u6570\u306e\u5b9a\u6570 \\(m\\) \u3092\u7528\u3044\u3066 \\(f(x) = ( x+m )^3\\) \u3068\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4e09\u89d2\u4e0d\u7b49\u5f0f, \\(|x| \\geqq 1\\) \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n| px+q | \\leqq |px| +|q| \\leqq |p|x +|q|x = \\left( |p|+|q| \\right) x\n\\]\r\n\u306a\u306e\u3067, \\(r = |p|+|q|\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n| px+q | \\leqq rx\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(\\alpha -\\beta = \\dfrac{\\alpha^3 -\\beta^3}{\\alpha^2 +\\alpha \\beta +\\beta^2}\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\left| \\sqrt[3]{f(x)} -x -k \\right| = \\left| \\dfrac{f(x) -(x+k)^3}{\\left\\{ f(x) \\right\\}^{\\frac{2}{3}} +(x+k) \\left\\{ f(x) \\right\\}^{\\frac{1}{3}} +(x+k)^2} \\right| \\quad ... [1]\n\\]\r\n\u3053\u3053\u3067, \\(k=\\dfrac{a}{3}\\) \u3068\u304a\u3051\u3070, [1]\u306e\u5206\u5b50\u306f\r\n\\[\\begin{align}\r\nf(x) -(x+k)^3 & = x^3 +ax^2 +bx +c -\\left( x+\\dfrac{a}{3} \\right)^3 \\\\\r\n& = \\left( b -\\dfrac{a^2}{9} \\right) x +c -\\dfrac{a^3}{27}\n\\end{align}\\]\r\n\u307e\u305f, \\(a , b , c\\) \u306f\u3059\u3079\u3066\u6b63\u306a\u306e\u3067\r\n\\[\r\n\\left\\{ f(x) \\right\\}^{\\frac{1}{3}} \\geqq x , \\ x+k = x+\\dfrac{a}{3} \\geqq x\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u306e\u5206\u6bcd\u306f\r\n\\[\r\n\\left\\{ f(x) \\right\\}^{\\frac{2}{3}} +(x+k) \\left\\{ f(x) \\right\\}^{\\frac{1}{3}} +(x+k)^2 \\geqq 3x^2 \\geqq x^2\n\\]\r\n\u4ee5\u4e0a\u3092\u7528\u3044\u308c\u3070, \\(p= b -\\dfrac{a^2}{9}\\) , \\(q = c -\\dfrac{a^3}{27}\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n[1] \\leqq \\left| \\dfrac{px+q}{x^2} \\right| \\quad ... [1]'\n\\]\r\n (1)<\/strong> \u306e\u7d50\u679c\u3092\u4e21\u8fba \\(x^2\\) \u3067\u5272\u308c\u3070\r\n\\[\r\n\\left| \\dfrac{px+q}{x^2} \\right| \\leqq \\dfrac{r}{x}\n\\]\r\n\u3092\u6e80\u305f\u3059\u5b9a\u6570 \\(r\\) \u304c\u5b58\u5728\u3059\u308b\u306e\u3067, \\(l=r\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n[1]' \\leqq \\dfrac{l}{x}\n\\]\r\n\u3068\u306a\u308b \\(l\\) \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\r\n (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3068, \\(\\sqrt[3]{f(x)} \\geqq x+\\dfrac{a}{3}\\) \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n0 \\leqq \\sqrt[3]{f(x)} -x -\\dfrac{a}{3} \\leqq \\dfrac{l}{x} \\quad ... [2]\n\\]\r\n\u3092\u6e80\u305f\u3059\u5b9a\u6570 \\(l\\) \u304c\u5b58\u5728\u3059\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u81ea\u7136\u6570 \\(n\\) \u3092 \\(x\\) \u306b\u4ee3\u5165\u3059\u308b\u3068,\r\n\\[\r\n0 \\leqq \\sqrt[3]{f(n)} -n -\\dfrac{a}{3} \\leqq \\dfrac{l}{n}\n\\]\r\n\u3053\u3053\u3067, \u4e21\u8fba \\(n \\rightarrow \\infty\\) \u306e\u3068\u304d\u3092\u8003\u3048\u308b\u3068, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{l}{n} =0\\) \u306a\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u304b\u3089\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\sqrt[3]{f(n)} -n -\\dfrac{a}{3} \\right) & =0 \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\sqrt[3]{f(n)} -n \\right) & = \\dfrac{a}{3}\n\\end{align}\\]\r\n\u6761\u4ef6\u3088\u308a, \u5de6\u8fba\u306f\u81ea\u7136\u6570\u306a\u306e\u3067, \\(\\dfrac{a}{3}\\) \u3082\u81ea\u7136\u6570\u3068\u306a\u308b.
\r\n[2] \u306b\u304a\u3044\u3066, \\(l\\) \u3088\u308a\u5927\u304d\u3044 \\(4\\) \u3064\u306e\u81ea\u7136\u6570 \\(n _ k \\ ( k=1, 2, 3, 4 )\\) \u3092\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\sqrt[3]{f(n _ k)} -n _ k & -\\dfrac{a}{3} = 0 \\\\\r\n\\text{\u2234} \\quad f(n _ k) & = \\left( n _ k+\\dfrac{a}{3} \\right)^3\n\\end{align}\\]\r\n\\(f(n)\\) \u306f \\(3\\) \u3064\u306e\u5b9a\u6570\u3092\u3082\u3064\u306e\u3067, \\(4\\) \u3064\u306e\u5909\u6570\u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u6052\u7b49\u5f0f\r\n\\[\r\nf(n) = \\left( n+\\dfrac{a}{3} \\right)^3\n\\]\r\n\u306f\u4efb\u610f\u306e\u5b9f\u6570\u306b\u5bfe\u3057\u3066\u6210\u7acb\u3059\u308b.
\r\n\u3088\u3063\u3066, \u81ea\u7136\u6570 \\(m = \\dfrac{a}{3}\\) \u3068\u304a\u304f\u3053\u3068\u3067, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(a , b , c\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3057, \\(x\\) \u306e\u95a2\u6570 \\(y = x^3 +ax^2 +bx +c\\) \u3092\u8003\u3048\u308b. \u4ee5\u4e0b, \u5b9a\u6570\u306f\u3059\u3079\u3066\u5b9f\u6570\u3068\u3059\u308b. (1)\u3000\u5b9a\u6570 \\(p , q\\) \u306b\u5bfe\u3057, \u6b21\u3092\u307f\u305f\u3059\u5b9a\u6570 … \u7d9a\u304d\u3092\u8aad\u3080