Proof of the Second Partial Derivative Test for functions of 2 Variables

Abstract

Analyzing polynomials of two variables and using Taylor series, will show the significance of the second partial derivative of a polynomial. This proof will show the importance of Hessian Matrices, Determinants, partial derivatives and Linear Algebra, in hopes of showing continuity, and differentiability. Programs such as Maple will also aid in proving how these second partial derivatives relate to the original function, while providing a visual aid to the audience. This proof will provide eveidence of the second partial derivative test, and use complete, correct mathematically rigorous language to express inutitive ideas behind the mathematical elements that are employed.