\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}