From the definition of convex function of many variables and the conclusion in reference,we can get some necessary and sufficient conditions for judging the convexity of function of many variables by directional derivative and limit.

This article presents the definition of binary convex function,proves that the binary convex function in open and convex region must be continuous,popularizes Jensen s inequality and establishes the decision theorem of binary convex function.

In this paper,we discussed the the extremum of function of several variables,and generalized the result of paper[2],Then we give the method of solving the extrme value of n-component function using first class partial derivative.

The teaching of function of several variables is a difficult point in higher mathematics teaching.

The sufficient condition of the extreme value of function of many variables;

Seeking the maximum and minimum of function is the perpetual topic in the medium-sized mathematics,and the function of many variables are difficulties.

Then, the approach of establishing rock strength theory using the method by which the extremum is obtained for pluralistic function is discussed according to the concept of yielding or failure of rock and a rock damage statistical strength theory.

The differential calculus of the pluralistic function is both the key and difficult point in higher mathematics.

The sufficient conditions for extreme of multivariate functions;

In this paper, a convenient judgement method about extreme value of one class multivariate functions—trivariate functions was given, and its application in reality was discussed.

The extreme conditions for the monovariate functions were applied to the multivariate functions and then an effective method to decide the extreme values for the multivariate functions was presented.