\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\medskip \begin{alertblock}{} The reverse statement is not true! Having $f'(c) = 0$ does not guarantee that $f(c)$ is a minimum or maximum. \end{alertblock} \pause\bigskip \begin{minipage}{.4\textwidth} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2}{2}{-2.5}{2.5}{1} \diagramannotatey{-1,1,2} \diagramannotatex{-1,1} \diagramannotatez \begin{scope}[cgreen,ultra thick] \draw plot[smooth,domain=-1.4:1.4,samples=30] function{x**3}; \end{scope} \begin{scope}[very thick] \node[cgreen,include] (a) at (0,0) {}; \draw[cred] ($(a)+(-1cm,0)$) -- ($(a)+(1cm,0)$); \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{minipage}{.59\textwidth} For example: \begin{talign} f(x) = x^3 \end{talign} \pause Then $f'(0) = 0$.\\ \pause \alert{But there is no minimum or maximum.} \end{minipage} \vspace{10cm} \end{frame}