问题标题:
f(x)在x=0处可导,有F(x)=f(x)(1+|sinx|),则证明F(x)在x=0处可导的充要条件是f(0)=0
问题描述:
f(x)在x=0处可导,有F(x)=f(x)(1+|sinx|),则证明F(x)在x=0处可导的充要条件是f(0)=0
乔宇回答:
F(x)=f(x)(1+|sinx|),F(0)=f(0)
F'(0)=lim(x->0)[F(x)-F(0)]/x
=lim(x->0)[f(x)*(1+|sinx|)-f(0)]/x
=lim(x->0)[f(x)-f(0)]/x+lim(x->0)f(x)*|sinx|/x
=f'(0)+lim(x->0)f(x)*|sinx|/x
lim(x->0+)|sinx|/x=1,lim(x->0-)|sinx|/x=-1
于是lim(x->0)f(x)*|sinx|/x存在lim(x->0)f(x)=0
f(x)在x=0处可导,必连续,故lim(x->0)f(x)=f(0)=0
即F(x)在x=0处可导的充要条件是f(0)=0.
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