最值理论中一个结论的证明

A Theorem about Global Extreme Value

同济大学《高等数学》最值理论中有结论称"若一个函数在一个区间内可导且只有一个驻点,并且这个驻点是函数的极值点,那么此驻点也是该函数的最值点",但并未给出证明.学生们对此频感好奇.受Fermat引理启发,利用反证法可获一个比此结论更为一般的定理.

“If a differentiable function on an interval has only one stationary point ,and the stationary point is a local extremum point of the function ,then the stationary point is also the global extremum point of the function”is a statement appeared ,without proof ,in the textbook of Higher Mathematics published by Tongji university . Students are usually curious about how to prove the result .In this paper ,inspired by Fermat’s lemma ,we establish a theorem ,which is more general than the result above .