Finding extreme values of a multivariable function

The general idea is analogous to single variable functions. Whether we are discussing the function's domain or a subdomain of it, we have to use derivatives to analyse how the function behaves to know whether a point is a maximum or a minimum. For two variables the idea is the same as for one variable. In a certain interval the function can be constant, crescent or decrescent. For three and more variables we lose the function's graph, but the algebra is the same. In case the function is strictly crescent or strictly decrescent, there isn't a maximum or a minimum unless we define a closed interval.

Let [math]\displaystyle{ P }[/math] be a point of many coordinates and [math]\displaystyle{ f }[/math] a multivariable function with a domain [math]\displaystyle{ D }[/math]. [math]\displaystyle{ P_0 \in D }[/math] is a point of (absolute or global) maximum of [math]\displaystyle{ f }[/math] if [math]\displaystyle{ f(P) \leq f(P_0) }[/math] for all [math]\displaystyle{ P \in D }[/math]; and [math]\displaystyle{ P_0 }[/math] is a point of (absolute or global) minimum if [math]\displaystyle{ f(P) \geq f(P_0) }[/math] for all [math]\displaystyle{ P \in D }[/math].

Let [math]\displaystyle{ f }[/math] be a continuous function over a compact space. Then [math]\displaystyle{ f }[/math] has at least one point of maximum and one point of minimum.

The proof is beyond the reach of Calculus. The idea is the same for single variable functions. With the function being continuous and the set of points in which its defined being a closed space, we can guarantee that somewhere in that space the function has a point of maximum and a point of minimum. Excluding the case of a constant function, the functions that we are considering do not have discontinuities such as a limit approaching infinity.

This theorem only states that the points exists. It doesn't say anything about their location or how to find them.

Compact space: this term comes from topology. It's a synonym for a closed and limited set. For functions of two or more variables it's the equivalent of a closed interval for functions of one variable.