The Extreme Value Theorem for Functions of Several Variables

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.