单峰函数最值定理的推广

An extension of the maximum and minimum theorem of unimodal function

单峰函数最值定理广泛应用于最值问题中,但它要求驻点惟一.本文讨论了多驻点情形下的最值问题,给出2个主要结论:(1)若可导函数f在某一区间内的所有驻点组成的集合是孤立点集,且函数f的极大点个数与极小点个数不相等时,则函数f在该区间上存在最值.(2)若可导函数f在某一区间内存在最小驻点和最大驻点,且这两个驻点均为极大(小)点时,则函数f在该区间上存在最大(小)值.

The maximum and minimum theorem of unimodal function is used in the problems of maximum and minimum generally, but the number of stationary points is limited only one. Maximum and minimum of a function when it has multiple stationary points is discussed, and two main results are obtained： （ 1 ） If the set of stationary points of a differentiable function f on an interval is isolated, and the number of maximum points and the number of minimum points off are not equal, then f has maximum or minimum on that interval. （2） If a differentiable function f has maximum stationary point and minimum stationary point on an interval, and the two points both are maximum point （or minimum point）, thenfhas maximum （or minimum） on that interval.