This chapter explores how to find the extreme values of a function of two variables. The tangent plane attached to the surface which represents the function of two variables is a way to approximate the function near the point of attachment. The flatness of the tangent plane occurs at the minimum and maximum of the function. The chapter describes nonzero values and continuity, saddle points, and two dimensional extreme test including saddle points. It discusses symmetric problems using positive and definite matrices. A canonical form for a symmetric matrix is reviewed along with signed definite matrices. The chapter provides a deeper look at the extremals and the second order tests for extremals for functions of two variables.