Maximum profit

Text

A firm sells $x$ units at price $p(x)=100-x$ (euros), with total cost $C(x)=200+20x$. Find the quantity maximizing the profit $P(x)=x\,p(x)-C(x)$ and the maximum profit.

Solution

Revenue $R=100x-x^{2}$; profit $P=-x^{2}+80x-200$. $P'(x)=-2x+80=0\Rightarrow x=40$; $P''<0$: maximum. $P(40)=1400$ euros. $\boxed{x=40;\quad P_{\max}=1400\ \text{euros}}$