Let R(x) be the revenue and C(x) be the cost from...

Let $R(x)$ be the revenue and $C(x)$ be the cost from manufacturing $x$ items. Profit is defined as $P(x)=R(x)-C(x) .$ Show that at the value of $x$ that maximizes profit, marginal revenue equals marginal cost.

Key Concepts

Marginal Cost

Marginal cost is the derivative of the cost function with respect to the number of items produced. It indicates the extra cost incurred from producing one additional unit. In production and economic decision-making, it is a critical measure to assess how costs evolve with changes in production levels.

First-Order Condition for Optimization

In calculus-based optimization, the first-order condition requires that the derivative of the profit function be zero at a maximum or minimum. For the profit function, this means setting the derivative, which is the difference between marginal revenue and marginal cost, equal to zero. This leads directly to the condition that marginal revenue equals marginal cost at the profit maximum, making it a cornerstone principle in economic optimization.

Profit Function

The profit function is defined as the difference between the revenue function and the cost function. It measures the net gain or loss for a business when producing and selling a certain number of goods. This concept is fundamental in microeconomics and business analysis, as it provides a basis for determining production levels that yield the highest financial returns.

Marginal Revenue

Marginal revenue is the derivative of the revenue function with respect to the number of items produced or sold. It represents the additional income earned by selling one more unit of a product. This concept is essential in economic analysis for understanding how revenue changes as production levels vary.

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