\begin{frame} \frametitle{Exam Task from 2003} \vspace{-1ex} \begin{exampleblock}{} \begin{minipage}{.54\textwidth} Find the area of the largest rectangle that can be inscribed as shown in the triangle. \end{minipage} \begin{minipage}{.45\textwidth} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=1cm,scale=.9] \diagram{-.5}{5}{-.5}{3.5}{1} \diagramannotatez \begin{scope}[ultra thick] %\draw plot[smooth,domain=-.5:4,samples=30] function{ (x**3 - 3*x**2 + 1)/5}; \draw[black] (0,3) -- node[at start,left] {$(0,3)$} node[at end,below] {$(4,0)$} (4,0); \draw[cred,fill=cred!20] (0,0) rectangle (2,1.5); \mpause[1]{ \node[below] at (2,0) {$x$}; } \end{scope} \end{tikzpicture} } \end{center} \end{minipage} \pause\smallskip The line trough $(0,3)$ \& $(4,0)$ has the equation: \alert{$\ell(x) = \pause -\frac{3}{4}x + 3$}\hspace{-2ex} \pause\medskip The area $A$ of the rectangle depends on the width $x$:\vspace{-1ex} \begin{talign} A(x) &= \mpause[1]{x \cdot \ell(x)} \mpause[2]{ = x \cdot (-\frac{3}{4}x + 3)} \mpause[3]{ = -\frac{3}{4}x^2 + 3x} \quad \mpause[4]{\text{for $x$ in $[\mpause[5]{0},\mpause[5]{4}]$}} \\ \mpause[6]{A'(x) }&\mpause[7]{= }\mpause[8]{-\frac{3}{2}x + 3} \quad\quad \mpause[9]{A'(x) =0 \;\iff\;}\mpause[10]{\frac{3}{2}x = 3}\mpause[11]{\;\iff\; x = 2} \end{talign} \pause\pause\pause\pause\pause\pause\pause\pause\pause\pause\pause\pause Thus the only critical number is $2$. The value of $A(x)$ at $0$, $2$, $4$: \begin{talign} A(0) = \mpause[1]{0} && \mpause[2]{A(2) =} \mpause[3]{3} && \mpause[4]{A(4) =} \mpause[5]{0} \end{talign} \pause\pause\pause\pause\pause\pause The the area of the largest rectangle is \alert{$3$}. \end{exampleblock} \end{frame}