The Extreme Value Theorem for Functions of Several Variables

Recall from the Determining Extreme Values of Functions of Several Variables page that if extreme values to a function of several variables $z = f(x_1, x_2, ..., x_n)$ exist, then all we need to do is check the critical points (points where $\nabla f(x_1, x_2, ..., x_n) = \vec{0}$), singular points (points where $\nabla f(x_1, x_2, ..., x_n)$ does not exist), and boundary points.

Also recall from The Extreme Value Theorem for single variable functions says that if a single variable function $f$ is continuous on the closed and bounded interval $[a, b]$ then $f$ attains an absolute maximum value and absolute minimum value of $[a, b]$. The following theorems are an extension of the Extreme Value Theorem for functions of several variables.

The Extreme Value Theorem for Functions of Two Variables

Theorem 1 (The Extreme Value Theorem for Functions of Two Variables): Let $z = f(x, y)$ be a continuous two variable real-valued function. If $D(f)$ is a closed and bounded set in $\mathbb{R}^2$ then $R(f)$ is a closed and bounded set in $\mathbb{R}$ and there exists $(a, b), (c, d) \in D(f)$ such that $f(a, b)$ is an absolute maximum value of $f$ and $f(c, d)$ is an absolute minimum value of $f$.

We will not prove Theorem 1 above. Instead, we will jump right into the strategy for applying theorem 1 provided that $z = f(x, y)$ is a continuous two variable function whose domain $D(f)$ is closed and bounded.

Method for Finding Absolute Extrema for Two Variable Functions with Closed and Bounded Domains: Let $z = f(x, y)$ be a continuous two variable real-valued function whose domain $D(f)$ is closed and bounded. To compute the absolute maximum and absolute minimum values guaranteed by theorem 1: 1) Verify that that domain of $f$ is indeed closed and bounded. 2) Find the critical points of $f$, that is, points for which $\nabla f(x, y) = (0, 0)$. Then compute the values of the critical points of $f$. 3) Determine the extreme values along the boundary of $f$. This is possible provided that the domain is closed and bounded as verified by step 1. 4) Analyze the values found in steps 2 and 3. The largest value is the absolute maximum of $f$, and the smallest value is the absolute minimum of $f$.

The Extreme Value Theorem for Functions of More Than 2 Variables

Theorem 2 (The Extreme Value Theorem for Functions of $n$ Variables): Let $z = f(x_1, x_2, ..., x_n)$ be a continuous $n$ variable real-valued function. If $D(f)$ is a closed and bounded set in $\mathbb{R}^n$ then $R(f)$ is a closed and bounded set in $\mathbb{R}$ and there exists $(x_1, x_2, ..., x_n), (y_1, y_2, ..., y_n) \in D(f)$ such that $f(x_1, x_2, ..., x_n)$ is an absolute maximum value of $f$ and $f(y_1, y_2, ..., y_n)$ is an absolute minimum value of $f$.

The strategy given above for two variable functions can easily be extended to find absolute maximum and absolute minimum values of a function $f$ of several variables provided that $f$ is continuous and that $D(f)$ is a closed and bounded set.