作者Sfly (topos)
看板Math
标题Re: [微积] sin x^2 sin^2 x 比大小
时间Sun Feb 13 18:27:53 2011
※ 引述《GSXSP (Gloria)》之铭言：
: 比大小:
: sin x^2 , sin^2 x for 0 < x < 1
Let f(x) satisfy the following condition
i) f(0)=0
ii) x> f(x)>0 on (0,1)
iii) f'(x)>0 decreases on (0,1).
Then f(x^2) > f(x)^2 on (0,1).
pf.
g(x):= f(x^2)-f(x)^2
g'(x)=2xf'(x^2) - 2f(x)f'(x)
Hence g(0)=g'(0)=0 by i),
and g'(x) > 0 on (0,1) by ii) & iii)
Therefore, g(x)>0 on (0,1).
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※ 编辑: Sfly 来自: 131.215.6.212 (02/13 18:28)