作者LuisSantos (^______^)
看板trans_math
标题Re: [微分应用]
时间Mon Jun 18 12:53:35 2007
※ 引述《lukechen90 (...?)》之铭言：
: 1.show that f'''(x)>0 on an interval, then f is concave upward on that
: interval.
: 2.show that -1/2<x/(1+x^2)<1/2 for every value of x
: = =
: 这两题不知道从哪里切入算会比较好Q_Q
: 麻烦高手帮解答~~
x
2. 令 f(x) = ---------
1 + x^2
1 + x^2 - (2)(x^2) 1 - x^2 (1 - x)(1 + x)
则 f'(x) = -------------------- = --------- = ----------------
1 + x^2 1 + x^2 1 + x^2
令 f'(x) = 0 , 则 x = 1 或 -1
↘ | ↗ ↗ |↘
-------------------------
-1 1
-1
由上图得知 f(x) 在 x = -1 时 有极小值f(-1) = ---
2
1
在 x = 1 时 有极大值f(1) = ---
2
-1 x 1
所以 --- ≦ --------- ≦ ---
2 1 + x^2 2
--
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