多元函数极值和条件极值的一般判定方法

本文较为完整地探讨了多元函数极值和条件极值的一般判定方法和求法。通过研究多元微分与一元微分之间的关系,把多元函数的极值判定问题转化为二次型的正定、负定判定问题,或转化为一阶方向导函数是否变号的问题。对于条件极值,研究了适用于所有情况的降维求极法,比拉格朗日乘数法更加直观、计算简便,并且同时解决了条件极值的判定问题。

General Methods of Determining the Type of Extremum and Constrained Extremum of Multivariate Function

This article has completely discussed two general methods of determining the types of extrema and constrained extrema of multivariate functions.It has introduced a new way to work out their values,too.Through researching the relationship between one-variate differential and multivariate differential,the problems about extrema of multivariate function can be transformed as the questions of deciding positive definition or negative definition of quadric forms.Moreover,they can be also turned into determining whether the sign of a first order directional derivative function is changed from positive to negative or vice versa.With regard to the constrained extremum,it has studied a means called degrading dimensions to compute extremum values,which is suitable for all situations and is more intuitionistic and convenient than Lagrange Multipliers.Besides,it has solved the problem of determining the type of constrained multivariate extremum simultaneously.