Suppose that a person is going to sell Fizzy Cola at a football game and must decide in advance how much to order. Suppose that he makes a gain of $m$ cents on each quart that he sells at the game but suffers a loss of $c$ cents on each quart that he has ordered but does not sell. Assume that the demand for Fizzy Cola at the game, as measured in quarts, is a continuous random variable $X$ with PDF $f$ and CDF $F$, show that his expected profit will be maximized if he orders an amount $α$ such that $$F(α) = \frac{m}{m + c}$$
How do I go about proving this?
Think about it incrementally. Let $G$ be expected profit, and suppose he has already decided to order $X$ quarts. Now write the expected profit $dG$ from ordering an additional $dX$ quarts. That additional profit can be written as a function of $m$, $c$, and $F$. There comes a point where $dG=0$, and that's the point at which he wants to stop increasing the order.