\begin{frame} \frametitle{Derivatives and the Shape of a Graph} \begin{block}{} If $f'(x) > 0$ on an interval, then $f$ is increasing on that interval. \pause\medskip If $f'(x) < 0$ on an interval, then $f$ is decreasing on that interval. \end{block} \pause \begin{exampleblock}{} Where is $f(x) = 3x^4 - 4x^3 -12x^2 + 5$ increasing/decreasing? \pause \begin{talign} f'(x) = \mpause[1]{12x^3 - 12x^2 - 24x} \mpause[2]{= 12x(x-2)(x+1)} \end{talign} \pause\pause\pause {\small \begin{tabular}{|c|c|c|c|c|l|} \hline Interval & $12x$ & $x-2$ & $x+1$ & $f'(x)$ & \\ \hline \mpause[1]{x < -1} & \mpause[5]{-} & \mpause{-} & \mpause{-} & \mpause{-} & \mpause{decreasing on $(-\infty,-1)$} \\ \hline \mpause[2]{-1 < x < 0} & \mpause[10]{-} & \mpause{-} & \mpause{+} & \mpause{+} & \mpause{increasing on $(-1,0)$} \\ \hline \mpause[3]{0 < x < 2} & \mpause[15]{+} & \mpause{-} & \mpause{+} & \mpause{-} & \mpause{decreasing on $(0,2)$} \\ \hline \mpause[4]{2 < x} & \mpause[20]{+} & \mpause{+} & \mpause{+} & \mpause{+} & \mpause{increasing on $(2,\infty)$} \\ \hline \end{tabular} } \pause[31] \begin{center}\vspace{-1ex} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm] \diagram{-2}{4}{-1}{1.3}{1} \diagramannotatez \diagramannotateyy{-1/-30,1/30} \diagramannotatex{-2,-1,1,2,3} \begin{scope}[ultra thick] \draw[cgreen] plot[smooth,domain=-2:3,samples=30] function{(3*x**4 - 4*x**3 - 12*x**2 + 5)/30}; \end{scope} \end{tikzpicture}\vspace{-1ex} } \end{center} \end{exampleblock} \end{frame}