设f(x)有二阶连续导数且f'(x)=0,lim(x趋向于0)f''(x)/|x|=1则

问题描述:

设f(x)有二阶连续导数且f'(x)=0,lim(x趋向于0)f''(x)/|x|=1则

f(x) = (1/6)|x^3|
分析:
如果x>0, f(x) = (1/6)x^3, f'(0) = 0, f''(x) = x, and f''(x)/|x|=1 当x->0+.
如果x0-.
由此可见,f(x) = (1/6)|x^3| 满足题给所有条件.