16 - Higher Partials and Applications

Abstract

This chapter discusses higher partials and applications, mixed partials, Taylor approximation, stability, constrained optimization, and constraint problems. A function of two variables f(x, y) has two first partial derivatives, each itself a function of two variables. Each in turn has two first partial derivatives; these four new functions are the second derivatives of f(x, y). The chapter presents the development of second derivative tests for maxima and minima. Stability is a modern term for the behavior of a function near a critical point. It is assumed that all functions have continuous first, second, and third partial derivatives. The constraint problem in three variables is to maximize or minimize a function f(x, y, z) subject to a constraint g(x, y, z) =c.