若F(0)是偶函数,且F(0)的导数存在.证明:F(0)的导数是0

Question

如题所述

Answer 1

f（x）=f（-x）在x趋向于0的时候对式子两边同时求导则有f'（x）=-f'（x）化简并带入x=0则有f‘（0）=0

Answer 2

f'(0)
＝lim(x→0) [f(x)－f(0)]/x 替换x为-t
＝lim(t→0) [f(-t)－f(0)]/(-t)
＝lim(t→0) [f(t)－f(0)]/(-t)
＝－lim(t→0) [f(t)－f(0)]/t
＝－f'(0)
所以，f'(0)＝0本回答被提问者采纳