\begin{frame} \frametitle{Maximum and Minimum Values} \begin{block}{Fermat's Theorem} \alert{If} $f$ has a local maximum or minimum at $c$ and $f'(c)$ exists, \\ \alert{then} $f'(c) = 0$. \end{block} \pause\bigskip We can now rephrase the the theorem as follows: \begin{block}{} If $f$ has a local extremum at $c$, then $c$ is a critical number of $f$. \end{block} \pause\bigskip We can use this to look for global extrema on intervals: \begin{block}{Closed Interval Method} To find the \emph{absolute} maximum and minimum values of a \alert{continuous} function $f$ on an \alert{closed} interval $[a,b]$: \begin{enumerate} \pause \item Find the values of $f$ at critical numbers of $f$ in $(a,b)$. \pause \item Find the values of $f$ at the endpoints of the interval. \pause \item The largest value of (1) and (2) is the absolute maximum, the lowest the absolute minimum. \end{enumerate} \end{block} \vspace{10cm} \end{frame}