Question

数学高手帮我整一下这道证明题吧！！详细点哈呵呵谢啦

如果f(X)是偶函数，且f导（0）存在证明f导（0）=0

Answer 1

f导（0）存在 -->f'(0+0) = f'(0-0)

f'(0+0)
=lim(h->0+) [f(0+h)-f(0)]/h
=lim(h->0+) [f(h)-f(0)]/h

f'(0-0)
=lim(h->0+) [f(0-h)-f(0)]/(-h)
= -1*lim(h->0+) [f(h)-f(0)]/h
= -f'(0+0)

∴f'(0)=f'(0-0)= -f'(0+0) = 0