Question

一道求函数可导条件的题目设f(x)可导,F(x)=f(x)(1+｜sinx｜）,若使F（x)在x=0处可导,则必有---

Answer 1

左右极限=0,F(x)=f(x)(1+sinx)=f(0)(x→0+),F(x)=f(x)(1-sinx)=f(0)(x→0-)

左右导数=0,F'(x)=[F(x)-F(0)]/[x-0]=[f(x)(1+sinx)-f(0)]/x=f(x)cosx+(1+sinx)f'(x)=f(0)+f'(0){x→0+}

F'(x)=[F(x)-F(0)]/[x-0]=[f(x)(1-sinx)-f(0)]/x=-cosxf(x)+f'(x)(1-sinx)=-f(0)+f'(0){x→0-}

于是f(0)+f'(0)=-f(0)+f'(0),所以f(0)=0