3.1 Chapter Introduction
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All that is superfluous displeases God and Nature.
All that displeases God and Nature is evil.
—Dante Alighieri, circa 1300
\(\ldots\) namely, because the shape of the whole universe is most perfect, and, in fact, designed by the wisest creator, nothing in all of the world will occur in which no maximum or minimum rule is somehow shining forth.
—Leonhard Euler
In one-variable calculus, to test a function \(f\hbox{(}x\hbox{)}\) for a local maximum or minimum, we often use the second derivative. We look for critical points \(x_{0}\)—that is, points \(x_{0}\) for which \(f' \hbox{(}x_{0}\hbox{)} \,{=}\, 0\), and at each such point we check the sign of the second derivative \(f''(x_{0})\). If \(f''\hbox{(}x_{0}\hbox{)} \,<\, 0, f\hbox{(}x_{0}\hbox{)}\) is a local maximum of \(f\); if \(f''\hbox{(}x_{0}\hbox{)} \,>\, 0, f\hbox{(}x_{0}\hbox{)}\) is a local minimum of \(f\); if \(f''\hbox{(}x_{0}\hbox{)} \,{=}\, 0\), the test fails.
This chapter extends these methods to real-valued functions of several variables. We begin in Section 3.2 with a discussion of iterated and higher-order partial derivatives, and in Section 3.3 we discuss the multivariable form of Taylor’s theorem; this is then used in Section 3.4 to derive tests for maxima, minima, and saddle points. As with functions of one variable, such methods help us to visualize the shape of a graph.
In Section 3.5, we study the problem of maximizing a real-valued function subject to supplementary conditions, also referred to as constraints. For example, we might wish to maximize \(f\hbox{(}x, y, z\hbox{)}\) among those (\(x, y, z\)) constrained to lie on the unit sphere, \(x^{\,2}+y^{\,2}+z^{\,2} = 1\). Section 3.6 discusses a technical theorem (the implicit function theorem) useful for studying constraints. It will also be useful later in our study of surfaces.