A firm's revenue is $R(x) = 100x - x^{2}$ rupees and its cost is $C(x) = 20x + 300$ rupees for $x$ units. What output gives the greatest profit?

Explanation

1. 01

   Profit $P(x) = 80x - x^{2} - 300$, so $P'(x) = 80 - 2x$.

2. 02

   $P'(x) = 0$ at $x = 40$, and $P''(x) = -2 < 0$: a maximum.