设f(x)在x=0的某邻域内存在二阶导数,且f'(x)=0,lim(x→0)f''(x)/|x|=a,a>0,则A.f(0)是f(x)的极小值.B.f(0)是f(x)的极大值.C.在点(0,f(0))的左侧邻近,曲线y=f(x)是凹的,右侧邻近是凸的.D.在点(0,f(0))的左侧邻近,

Question

设f(x)在x=0的某邻域内存在二阶导数,且f'(x)=0,lim(x→0)f''(x)/|x|=a,a>0,则
A.f(0)是f(x)的极小值.
B.f(0)是f(x)的极大值.
C.在点(0,f(0))的左侧邻近,曲线y=f(x)是凹的,右侧邻近是凸的.
D.在点(0,f(0))的左侧邻近,曲线y=f(x)是凸的,右侧邻近是凹的.

Answer 1

lim(x→0)f''(x)/|x|=a,所以在x=0的某个小的邻域(-a,0)和(0,a)内,|x|>0,那么f''(x)>0.尽管f''(0)=0,但是在x=0的两侧,f''(x)是同号的,所以x=0不是拐点,所以c,d不对.由于f''(x)在(-a,a)内满足f''(x)>=0,所以(-a,a)内f'...