\begin{frame} \frametitle{Maximum and Minimum Values} \begin{minipage}{.5\textwidth} \begin{center} \scalebox{.6}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{6}{-.5}{4.5}{1} \diagramannotatey{1,2,3} \diagramannotatex{1,2,3,4,5} \diagramannotatez \begin{scope}[cgreen,ultra thick] %\draw plot[smooth,domain=-1:4,samples=200] function{(3*x**4 - 16*x**3 + 18*x**2)/15}; \draw (1,2) to[out=45,in=-95] (2,4); \draw (2,2) to[out=-70,in=170] (4,1); \node[include] at (1,2) {}; \node[exclude] at (2,4) {}; \node[include] at (2,2) {}; \node[include] at (4,1) {}; \end{scope} \end{tikzpicture} } \end{center} \end{minipage}~% \begin{minipage}{.49\textwidth} \pause Absolute minimum:\\\pause $f(4) = 1$ \pause\medskip Absolute maximum:\\\pause none \pause\medskip Not continuous on $[1,4]$! \end{minipage} \pause\medskip \begin{minipage}{.5\textwidth} \begin{center} \scalebox{.6}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-.5}{6}{-.5}{4.5}{1} \diagramannotatey{1,2,3} \diagramannotatex{1,2,3,4,5} \diagramannotatez \begin{scope}[cgreen,ultra thick] %\draw plot[smooth,domain=-1:4,samples=200] function{(3*x**4 - 16*x**3 + 18*x**2)/15}; \draw (1,1) to[out=45,in=-95] (2.8,4.5); \node[exclude] at (1,1) {}; \draw[gray,dashed] (3,0) -- (3,4.5); \end{scope} \end{tikzpicture} } \end{center} \end{minipage}~% \begin{minipage}{.49\textwidth} \pause Absolute minimum:\\\pause none \pause\medskip Absolute maximum:\\\pause none \pause\medskip Continuous on $(1,3)$, but this is not a closed interval! \end{minipage} \pause\smallskip \begin{alertblock}{} The function needs to be \emph{continuous} on a \emph{closed} interval $[a,b]$.\hspace{-10ex} \end{alertblock} \vspace{10cm} \end{frame}