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._数学解答

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.

人气：243 ℃　时间：2019-08-17 22:03:54

解答

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