\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 The theorem suggests where local extra can occur: \begin{itemize} \pause \item where $f'(c) = 0$, or \pause \item where $f'(c)$ does not exist. \end{itemize} \pause \begin{block}{} A \emph{critical number} of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$, or $f'(c)$ does not exist. \end{block} \pause \begin{exampleblock}{} What are the critical numbers of $f(x) = x^{3/5}(5-x)$? \pause \begin{talign} &f(x) = x^{3/5}(5-x) \mpause[1]{= 5x^{3/5} - x^{8/5}} \\ &\mpause[2]{f'(x) = } \mpause[3]{\frac{3}{x^{2/5}} - \frac{8}{5}x^{3/5}} \mpause[4]{= \frac{15}{5x^{2/5}} - \frac{8x}{5x^{2/5}}} \mpause[5]{= \frac{15 - 8x}{5x^{2/5}}} \end{talign} \pause\pause\pause\pause\pause\pause The critical numbers are $\frac{15}{8}$ ($f(c) = 0$) and $0$ ($f(c)$ does not exist)\hspace{-5ex} \end{exampleblock} \vspace{10cm} \end{frame}