\begin{frame}
  \frametitle{Optimization}

  \vspace{-1ex}
  \begin{exampleblock}{}
    A store sells 100 blu-ray players per week for $200$\$ each.
    A market survey shows that for each $10$\$ discount, the store would sell $40$ more players per week. 
    The store buys the players at a price of $150$\$ per piece.
    \pause\smallskip
    
    What selling price would maximize the profit of the store?
    \pause\smallskip

    \begin{overlayarea}{\textwidth}{5cm}
    \only<-22>{
    Introducing notation:
    \begin{itemize}
    \pause
      \item let $x$ be the discount
    \pause
      \item let $s$ be the number of players sold, and $p$ the profit
    \end{itemize}\vspace{-1ex}
    \pause
    \begin{talign}
      s(x) &= \mpause[1]{100 + 40\cdot \frac{x}{10}} \mpause{ = 100 + 4x}\\[-.5ex]
      \mpause{p(x) &= }\mpause{s(x) \cdot (200 - x - 150)} \mpause{ = (100 + 4x) \cdot (50-x)}\\[-.5ex]
      \mpause{&= -4x^2 + 100x + 5000} \hspace{1cm} \mpause{\text{for $x$ in $[\mpause[8]{0},\mpause[8]{50}]$}}\\[-.5ex]
      \mpause[9]{p'(x) &= -8x + 100} \hspace{1cm} \mpause{p'(x) = 0 \;\iff\; }\mpause{x=12.5}
    \end{talign}
    \pause\pause\pause\pause\pause\pause\pause\pause\pause\pause\pause\pause
    Note that $p(x)$ is continuous\pause, and\vspace{-1ex}
    \begin{talign}
      p(0) &= \mpause[1]{5000} &
      p(12.5) &= \mpause[2]{5625} &
      p(50) = \mpause[1]{0}
    \end{talign}
    \pause\pause\pause
    }
    \only<23>{
    \begin{center}
      \begin{tikzpicture}[default,scale=.65]
        \def\diaborderx{1.2cm}
        \def\diay{$p(x)$}
        \diagram{-0.5}{5}{-.5}{6}{1}
        \diagramannotatexx{1/10,2/20,3/30,4/40}
        \diagramannotateyy{1/1000,2/2000,3/3000,4/4000,5/5000}
        \diagramannotatez
        \begin{scope}[ultra thick]
          \draw[cgreen] plot[smooth,domain=0:5,samples=100] function{(100 + 4*10*x) * (50-10*x)/1000};
        \end{scope}
      \end{tikzpicture}
    \end{center}
    }
    \end{overlayarea}
    
    By the Closed Interval Method, \alert{$12.5$\$ discount for maximal profit}.\hspace*{-10ex}
  \end{exampleblock}
\end{frame}