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Students in a calculus class were asked to solve the following problem:

\begin{quote}
    A company is manufacturing widgets and selling them. Based on their current setups, they can only produce between 0 and 10,000 widgets per day. If they produce $x$ thousand widgets, the price they can sell them at is $p(x) = 20x - x^2 + 30$. However, the cost to produce $x$ widgets is $C(x) = 5x^2 + 105x + 10$. Find the production level that maximizes profit. 
\end{quote}

A student's work on this problem is as follows. 

\begin{quote}
    We know that the formula for profit is
    \[ P(x) = xp(x) - C(x) = 20x^2 - x^3 + 30x - (5x^2 - 105x + 10) = -x^3 + 15x^2 -75x - 10 \]
    We then take the derivative and set it to zero:
    \[ 0 = P'(x) = -3x^2 + 30 x - 75 = -3(x^2 + 10x-25) = -3(x-5)^2 \]
    Thus our only critical point is $x=5$.  This must be the point that maximizes profit, so the desired production level is 5,000 widgets.
\end{quote}

\begin{enumerate}
    \item There is a significant step at the end of the process that is missing. What is it?
    \item Sketch a graph of the profit function over the range $0$ to $10$. What happens at the critical point $x=5$?
    \item Work out a full solution to the problem. What should the company do?
\end{enumerate}

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