设f(x)在x=0处连续,且lim(x趋于0)f(x)/x^2=1 ,证明函数f(x)在x=0处可导且取得极小值.

问题描述:

设f(x)在x=0处连续,且lim(x趋于0)f(x)/x^2=1 ,证明函数f(x)在x=0处可导且取得极小值.

1、f(0)=lim f(x)=lim f(x)/x^2 *lim x^2=1*0=0,
于是f'(0)=lim [f(x)-f(0)]/x
=lim f(x)/x^2*x
=lim f(x)/x^2 *lim x
=1*0=0,
即f'(0)=0.
2、对e=1/2,存在d>0,使得
0