How to Calculate Maximum Profit in a Monopoly
Profit is maximized at the quantity of output where marginal revenue equals marginal cost. Marginal revenue represents the change in total revenue associated with an additional unit of output, and marginal cost is the change in total cost for an additional unit of output.
Therefore, both marginal revenue and marginal cost represent derivatives of the total revenue and total cost functions, respectively. You can use calculus to determine marginal revenue and marginal cost; setting them equal to one another maximizes total profit.
The monopolist’s demand curve
generated the total revenue equation.
Also assume your total cost equation is
Given these equations, the profitmaximizing quantity of output is determined through the following steps:

Determine marginal revenue by taking the derivative of total revenue with respect to quantity.

Determine marginal cost by taking the derivative of total cost with respect to quantity.

Set marginal revenue equal to marginal cost and solve for q.

Substituting 2,000 for q in the demand equation enables you to determine price.
Thus, the profitmaximizing quantity is 2,000 units and the price is $40 per unit.
The profitmaximizing quantity and price are the same whether you maximize the difference between total revenue and total cost or set marginal revenue equal to marginal cost.