Question

f(x)在x=0处可导，有F(x)=f(x)(1+|sin x|)，则证明F(x)在x=0处可导的充要条件是f(0)=0

Answer 1

F(x)=f(x)(1+|sin x|)，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。