多元函数极值的一些求法

Some Methods of Finding Extremums of Multivariate Functions

摘要: 函数的极值是函数性态的一个重要特征。本文先利用偏导数给出了一个二元函数极值判定的充分条件。其次，先给出方向导数的定义然后展示了利用方向导数判定二元函数极值的两个充分条件。同时，本文也探讨了利用拉格朗日乘数法研究多元函数的极值。简而言之，本文利用偏导数、方向导数和拉格朗日乘数法总结了多元函数极值的一些求法，这为进一步研究多元函数的最值奠定了基础。

Abstract: The extremum of a function is an important characteristic of its properties. Firstly, a sufficient condition for extremums of a binary function is given by using partial derivatives. Secondly, the definition of the directional derivative is given, and then two sufficient conditions for extremums of binary functions by using directional derivative are presented. Meanwhile, the extremums of multivariate functions by using the Lagrange multiplier method are discussed. In short, some methods of finding extremums of multivariate functions by using partial derivatives, directional derivatives and the Lagrange multiplier method are summarized, which lay a foundation for further research on the maximum value of multivariate functions.