This chapter focuses on the identification of the maximum, minimum, or saddle points located in the domain of the real-valued function in a graph of a function. The first part of this chapter focuses on the open set domain of a function and the second part on the closed set domain. The characterization of these points called critical points" is resolved with the Hessian matrix and the Bordered Hessian matrix. Finally, we will review the Implicit Function Theorem.
Keywords: Bordered Hessian matrix, Closed set, Critical points, Hessian matrix, Implicit function theorem, Inverse function theorem, Inflection value, Inverse function, Maximum points, Maximum value, Minimum points, Minimum value, Open set, Saddle point, Second order partial derivative.