\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}{} A local minimum/maximum does not guarantee that $f'(c)$ exists. \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=-2:2,samples=30] function{abs(x)}; \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \begin{minipage}{.59\textwidth} For example: \begin{talign} f(x) = |x| \end{talign} \pause Then $f(0) = 0$ is a local minimum.\\ \pause \alert{But $f'(0)$ does not exist.} \end{minipage} \pause \begin{alertblock}{} Care needed for applying the theorem (check both conditions)! \end{alertblock} \vspace{10cm} \end{frame}