【考研数学】设f(x)可导,F(x)=f(x)(1+|sin x|)则f(0)=0是F(x)在x=0处可导的（ ）条件如题,A.充要 B.充分非必要 C.必要非充分 D.非充分非必要
2019-04-18
【考研数学】设f(x)可导,F(x)=f(x)(1+|sin x|)则f(0)=0是F(x)在x=0处可导的（ ）条件
如题,A.充要 B.充分非必要 C.必要非充分 D.非充分非必要
优质解答
用导数的定义
当x趋向于正零时,F（x）在0处的导为：
lim (F(x)-F(0)) / x = lim (f(x) + f(x)sinx - f(0)) / x = lim (f(x) - f(0)) / x + lim f(x)sinx / x = f'(0) + f(0)
当x趋向于负零时,F（x）在0处的导为：
lim (F(x)-F(0)) / x = lim (f(x) - f(x)sinx - f(0)) / x = lim (f(x) - f(0)) / x - lim f(x)sinx / x = f'(0) - f(0)
F(x)在0处可导,则f'(0) + f(0) = f'(0) - f(0),f(0) = 0
用导数的定义
当x趋向于正零时,F（x）在0处的导为：
lim (F(x)-F(0)) / x = lim (f(x) + f(x)sinx - f(0)) / x = lim (f(x) - f(0)) / x + lim f(x)sinx / x = f'(0) + f(0)
当x趋向于负零时,F（x）在0处的导为：
lim (F(x)-F(0)) / x = lim (f(x) - f(x)sinx - f(0)) / x = lim (f(x) - f(0)) / x - lim f(x)sinx / x = f'(0) - f(0)
F(x)在0处可导,则f'(0) + f(0) = f'(0) - f(0),f(0) = 0