ABSTRACT
In this chapter, the authors optimize several variables’ functions over subsets S of Rn with nonempty interior. Many results are well known when dealing with functions of one variable (n = 1). The concept of differentiability offered useful and flexible tools to get local and global behaviors of a function. These results are generalized to functions of n variables in these notes. The authors obtain a characterization of local critical points as solutions of the vectorial equation. They use the second partial derivatives to identify the nature of the critical points and define the convexity-concavity property for a function of several variables, and show how to use it to identify the global extreme points. Finally, the authors extend the extreme value theorem to continuous functions on closed bounded subsets of Rn.
