\begin{frame} \frametitle{Maximum and Minimum Values} \vspace{-1ex} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{10}{-.5}{3}{1} \diagramannotatey{1,2} \diagramannotatex{1,2,3,4,5,6,7,8,9} \diagramannotatez \begin{scope}[ultra thick] %\draw[cgreen,ultra thick] plot[smooth,domain=-3:6,samples=200] function{sqrt(x+3)} node[below,xshift=-2mm,yshift=-2mm] {$\sqrt{x+3}$}; \begin{scope}[cgreen,ultra thick] \draw (.2,3) to[out=-45,in=180] (2,1.5) to[out=0,in=180] (4,2.5) to[out=0,in=180] (7,.5) to[out=0,in=-120] (9,3); \end{scope} \mpause[3]{ \node[include=cred] (a) at (7,.51) {}; \node[] (b) at ($(a)+(0,2)$) {absolute minimum}; \mpause[11]{ \node[] at ($(a)+(0,2.4)$) {local minimum \&}; } \draw[->,shorten >= 2mm] (b) -- (a); } \mpause[9]{ \node[include=cred] (a) at (4,2.5) {}; \node[] (b) at ($(a)+(0,-1.5)$) {local maximum}; \draw[->,shorten >= 2mm] (b) -- (a); } \mpause[10]{ \node[include=cred] (a) at (2,1.5) {}; \node[] (b) at ($(a)+(0,-1)$) {local minimum}; \draw[->,shorten >= 2mm] (b) -- (a); } \end{scope} \end{tikzpicture} } \end{center}\vspace{-1ex} \begin{block}{} Let $c$ be in the domain $D$ of $f$. Then $f(c)$ is the \begin{itemize} \pause \item \emph{absolute maximum} value of $f$ if $f(c) \ge f(x)$ for all $x$ in $D$ \pause \item \emph{absolute minimum} value of $f$ if $f(c) \le f(x)$ for all $x$ in $D$ \end{itemize} \end{block} \pause\pause Often called \emph{global maximum} or \emph{global minimum}.\\ \pause Minima and maxima are called \emph{extreme values} of $f$. \pause\medskip \begin{block}{} The number $f(c)$ is a \begin{itemize} \pause \item \emph{local maximum} value of $f$ if $f(c) \ge f(x)$ when $x$ is near $c$ \pause \item \emph{local minimum} value of $f$ if $f(c) \le f(x)$ when $x$ is near $c$ \end{itemize} \end{block} \vspace{10cm} \end{frame}