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\begin{center}

\begin{figure}[htbp]

\fbox{\resizebox{5.2in}{!}{\includegraphics{../images/instructors_guide_illustrations/max_min_on_domain}}}

\caption[Conditions for critical points on a bounded domain]{Critical points occur when any derivative is $0$ on the interior of the surface, its boundary, or the boundary's boundary.}

\label{fig:max_min_with_bounded_domain}

\end{figure}

\end{center}

The critical points of a function $f$ of $n$ variables occur where $\vec{\nabla} f = 0$. If the domain of $f$ is bounded, as shown in Figure \ref{fig:max_min_with_bounded_domain}, this boundary must also be checked for critical points. The boundary, which is a function of $n-1$ free variables, also has a boundary which must be checked for critical points. This process continues recursively until reaching the case of $0$ free variables, which reduces to just points. This process is an extension of the process learned in single variable calculus for finding maxima and minima for a function on a bounded interval.

Figure \ref{fig:max_min_with_bounded_domain} illustrates this process. The function $f$, a function of two variables, is locally flat where $\vec{\nabla} f = 0$. These points are indicated by the dashed blue circles. The boundary of the function consists of four edges. While $f$ was a function of two variables, each edge is defined by a relationship between the two variables, say $y$ and $x$. When each condition is substituted into the function $f$, the resulting function is restricted to this edge and is only a function of one variable. The orange dashed circles indicate the locations of the critical points along each edge. Finally, the endpoints of each edge, marked with black circles, must also be included as critical points. The function $f$ can be evaluated at each of these critical points, and the maximum and minimum values of $f$ can then be selected from this list.

Note, too, that the critical points along the region's boundary could also be solved using methods for constrained optimization.