(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 mar- ginal 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.

(a) Show that if the profit $P ( x )$ is a maximum, then the mar-
ginal 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

First-Order Condition for Profit Maximization

The first-order condition for profit maximization states that the derivative of the profit function with respect to quantity should be zero at the optimal production level. This implies that marginal revenue equals marginal cost, which is the point where any further change in production would not increase profit, thereby defining the condition for maximum profit.

Marginal Cost

Marginal cost is the derivative of the cost function with respect to quantity, signifying the additional expense incurred by producing one more unit. This concept is essential in economic decision-making, as it allows firms to evaluate the cost impact of increasing production.

Marginal Revenue

Marginal revenue is the derivative of the revenue function with respect to quantity, indicating the additional income gained from selling one more unit of a product. It is a crucial concept in marginal analysis, which helps in understanding how revenue changes with small variations in production levels.

Profit Function

The profit function represents the difference between total revenue and total cost. It is a fundamental concept in economics as it quantifies how much profit a firm makes when producing and selling a certain number of units. Understanding the profit function is key to determining the optimal production level that maximizes profit.

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