Profit-maximizing price for a product

Statement

For a product, demand is $D(p)=8000\left(1-\dfrac{p}{40}\right)$ and the unit cost is $6\ \text{€}$. Find the price that maximizes the profit $(p-6)\,D(p)$.

Solution

The profit is $\pi(p)=(p-6)\cdot 8000\left(1-\frac{p}{40}\right)=200\,(p-6)(40-p),$ a downward parabola vanishing at $p=6$ and $p=40$. The maximum is at the vertex, midway between the zeros: $p^{*}=\frac{6+40}{2}=\boxed{23\ \text{€}}.$ At that price $D(23)=8000\left(1-\tfrac{23}{40}\right)=3400$ units and the profit is $(23-6)\cdot 3400=57\,800\ \text{€}$.