\begin{frame} \frametitle{Derivatives of Basic Functions} \begin{exampleblock}{} Find the points of $f(x) = x^4 - 6x^2 + 4$ with horizontal tangent. \pause\bigskip Horizontal tangent means that the slope (the derivative) is $0$: \begin{talign} \frac{d}{dx} f(x) \;=\; \mpause[1]{4x^3 - 12x} \mpause[2]{\;=\; 4x(x^2 - 3)} \end{talign} \pause\pause\pause Thus $f'(x) = 0$ when $x = 0$ or $x = \sqrt{3}$ or $x = -\sqrt{3}$.\pause\medskip Thus the corresponding points are $(0,4)$, $(\sqrt{3},-5)$, $(-\sqrt{3},-5)$. \end{exampleblock} \pause \begin{center} \scalebox{.8}{ \begin{tikzpicture}[default,baseline=1cm,yscale=.4] \diagram{-4}{4}{-6}{5}{1} \diagramannotatez \begin{scope}[ultra thick] \draw[cred,dashed] (-1,4) -- node[above right] {$(0,4)$} (1,4) ; \draw[cred,dashed] ({-sqrt(3)-1},-5) -- ({-sqrt(3)+1},-5) node[below left] {$(-\sqrt{3},-5)$}; \draw[cred,dashed] ({sqrt(3)-1},-5) -- ({sqrt(3)+1},-5) node[below left] {$(\sqrt{3},-5)$}; \draw[cgreen,ultra thick] plot[smooth,domain=-2.5:2.5,samples=200] function{x**4 - 6*x**2 + 4}; \end{scope} \end{tikzpicture} } \end{center} \end{frame}