Methods of determining the extreme points for trivariate functions

There have been some researches in recent years, e.g. [2-5], on how to determine the extreme points of a smooth multivariate function, especially in the case where Hesse criterion is ineffective; however, most of the researches focus only on bivariate functions or lower-order stable points. In this essay we adopt Taylor series expansion of the function at the stable points, consider the positive definiteness of the corresponding homogeneous polynomial, raise the image criterion for determining the extreme points of a trivariate function, and finally extend the conclusions to higher-order situations. One fact is that this criterion is effective, and there's no need to consider the influence of the order of the stable points.