Final answer:
The number of units x that produce maximum revenue r is found where marginal revenue equals marginal cost (MR = MC). This is typically determined through calculus, finding when the derivative of revenue equals the derivative of cost. If MR never equals MC, there is no solution for maximizing revenue.
Step-by-step explanation:
Maximizing Revenue in Economics
To find the number of units x that produce maximum revenue r, one needs to understand the relationship between marginal revenue (MR) and marginal cost (MC). In economics, profit maximization occurs at the point where MR equals MC. This point can be found by performing a calculus operation, finding where the derivative of the revenue function is equal to the derivative of the cost function, signifying the point of no further revenue increase beyond this quantity of output.
To maximize profits, you should produce up to the quantity where marginal revenue is equal to marginal cost, MR = MC. Doing this ensures that every additional unit produced contributes as much as possible to total profit without increasing cost disproportionately. This is the point where producing more would result in increased costs that exceed the revenue generated by selling the additional units.
If there is no point where MR equals MC, or if the cost structure is such that costs always exceed revenues, then there is no solution for maximizing revenue as no positive profit space exists.