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In Economics, Profit Maximization is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem. The total revenue -- total cost method relies on the fact that profit equals revenue minus cost, and the marginal revenue -- marginal cost method is based on the fact that total profit in a perfectly competitive market reaches its maximum point where marginal revenue equals marginal cost.

Basic Definitions:

Any costs incurred by a firm may be classed into two groups: fixed cost and variable cost. Fixed costs are incurred by the business at any level of output, including zero output. These may include equipment maintenance, rent, wages, and general upkeep. Variable costs change with the level of output, increasing as more product is generated. Materials consumed during production often have the largest impact on this category. Fixed cost and variable cost, combined, equal total cost.

Revenue is the total amount of money that flows into the firm. This can be from any source, including product sales, government subsidies, venture capital and personal funds.

Marginal cost and revenue, depending on whether the calculus approach is taken or not, are defined as either the change in cost or revenue as each additional unit is produced, or the derivative of cost or revenue with respect to quantity output. It may also be defined as the addition to total cost as output increase by a single unit. For instance, taking the first definition, if it costs a firm 400 USD to produce 5 units and 480 USD to produce 6, the marginal cost of the sixth unit is approximately 80 dollars, although this is more accurately stated as the marginal cost of the 5.5th unit due to linear interpolation. Calculus is capable of providing more accurate answers if regression equations can be provided.
To obtain the profit maximizing output quantity, we start by recognizing that profit is equal to total revenue minus total cost. Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph. Finding the profit-maximizing output is as simple as finding the output at which profit reaches its maximum. That is represented by output Q in the diagram.
There are two graphical ways of determining that Q is optimal. Firstly, we see that the profit curve is at its maximum at this point (A). Secondly, we see that at the point (B) that the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), the surplus of revenue net of costs (B,C) is the greatest. Because total revenue minus total costs is equal to profit, the line segment C,B is equal in length to the line segment A,Q.
Computing the price at which to sell the product requires knowledge of the firm's demand curve. The price at which quantity demanded equals profit-maximizing output is the optimum price to sell the product