(a) Show that if the profit P(x) is a maximum, then the...

(a) Show that if the profit $P(x)$ is a maximum, then the
marginal revenue equals the marginal cost.
(b) If $C(x)=16,000+500 x-1.6 x^{2}+0.004 x^{3}$ is the
cost function and $p(x)=1700-7 x$ is the demand
function, find the production level that will maximize
profit.

Key Concepts

Demand and Cost Functions

The demand function relates the price of a product to the quantity demanded, and it is used to determine the revenue function by multiplying the price by the quantity sold. On the other hand, the cost function captures all the costs incurred in production as a function of quantity. Together, these functions form the basis for constructing the profit function and are essential components in determining the optimal level of production for maximum profit.

Marginal Revenue and Marginal Cost

Marginal revenue is the derivative of the revenue function, R(x), while marginal cost is the derivative of the cost function, C(x). The equality of these two marginal values at the maximum profit point is a fundamental condition in economics, encapsulating the idea that if producing one more unit would add more cost than revenue (or vice versa), then the current production level is optimal.

Marginal Analysis

Marginal analysis involves studying the effect of a small change in the level of production on the profit. Specifically, the derivative of the profit function, P'(x), indicates the additional profit from producing one more unit. A key insight is that at the profit-maximizing level where P(x) is at its maximum, the rate of change of the profit function is zero, meaning marginal revenue equals marginal cost.

Profit Function

The profit function is defined as the difference between revenue and cost, P(x) = R(x) ? C(x). It serves as a measure of how much profit is generated at different levels of production. In optimization, one is typically interested in finding the value of x that maximizes this function, thereby ensuring the most efficient use of resources and maximum gain.

Optimization Using Derivatives

The process of finding the maximum or minimum of a function is achieved by taking its derivative and setting it equal to zero to find critical points. In the context of profit maximization, taking the derivative of the profit function allows us to determine where the additional gain from producing one more unit (marginal revenue) is exactly offset by the additional cost of that unit (marginal cost), a necessary condition for optimization.

Transcript

Please give Ace some feedback