Example 3

Suppose a quantity x of a commodity can be produced at a total cost C(x) and sold for a total revenue of R(x), 0 < x < ∞. The profit is defined as the difference between the revenue and the cost,

P(x) = R(x) - C(x).

Show that if the profit has a maximum at x₀, then the marginal cost is equal to the marginal revenue at x₀,

R'(x₀) = C'(x₀).

In this problem it is understood that R(x) and C(x) are differentiate functions, so that the marginal cost and marginal revenue always exist. Therefore P'(x) exists and

P'(x) = R'(x) - C'(x).

Assume P(x) has a maximum at x₀. Since (0, ∞) has no endpoints and P'(x₀) exists, the Critical Point Theorem shows that P'(x₀) = 0. Thus

P'(x₀) = R'(x₀) - C'(x₀) = 0

and

R'(x₀) = C'(x₀).