Suppose the demand of a good is P=10−QP=10−Q

We derive the total revenue from the demand function:

• TR=P×QTR=(10−Q)×QTR=10Q−Q2TR=P×QTR=(10−Q)×QTR=10Q−Q2.

We now derive the marginal revenue from the total revenue function:

• MR=dTRdQMR=10−2QMR=dTRdQMR=10−2Q.

We derive the marginal cost from the total cost function:

• MC=dTCdQMC=2MC=dTCdQMC=2.

A profit-maximizing monopolist will produce up to the quantity where MR equals MC:

• 10−2Q=28=2QQ∗=410−2Q=28=2QQ∗=4

We plug Q=4Q=4 into the demand function to obtain the optimal price:

• P∗=10−4=$6P∗=10−4=$6

Hence, the monopolist's optimal price is $6, and the optimal quantity is 4 units.