Question

Find the profit-maximizing price.

Answer 1

Answer: $\ 11.5$

Step-by-step explanation:

Given

Demand function is $p=22-2q$

The average cost function is $c=2+q$

Total revenue is the product of demand and the price per unit.

$r=\left(22-2q\right)q$

Profit is given by the difference of the total revenue and the cost

$\Rightarrow P=r-c\\\Rightarrow P=22q-2q^2-2-q\\\Rightarrow P=-2q^2+21q-2$

Find the derivative of profit to get the maximum profit

$\Rightarrow P'=-4q+21\\\text{Put the derivative equal to 0 to get the maximum profit}\\\\\Rightarrow q=\dfrac{21}{4}$

Put q in the equation of demand to get the price

$\Rightarrow p=22-2\times \dfrac{21}{4}\\\\\Rightarrow p=22-10.5\\\Rightarrow p=\ 11.5$