Problem 46 - Calculus with Concepts in Calculus

Question

Suppose \(R(x), C(x)\), and \(P(x)\) denote the revenue, cost, and profit resulting from the manufacture and sale of \(x\) units of an item. Recall that $$ P(x)=R(x)-C(x) \quad \text { for } x \geq 0 $$ Assume that it is possible to make a maximum profit by manufacturing \(x_{0}\) units of the item. Show that if \(R\) and \(C\) are differentiable and \(x_{0}>0\), then \(R^{\prime}\left(x_{0}\right)=C^{\prime}\left(x_{0}\right)\) (that is, the marginal revenue at \(x_{0}\) equals the marginal cost at \(x_{0}\) ).

Step-by-Step Solution

Answer

At maximum profit, the marginal revenue equals the marginal cost: \( R'(x_0) = C'(x_0) \).

Step 1: Define Profit in Terms of Revenue and Cost

The profit function is given by combining the revenue and cost functions: \( P(x) = R(x) - C(x) \). This represents how profit changes with respect to the units \( x \) manufactured and sold.

Step 2: Differentiate the Profit Function

To find the rate of change of the profit with respect to \( x \), differentiate \( P(x) \) with respect to \( x \). Using the definition: \( P'(x) = R'(x) - C'(x) \). This represents the marginal profit.

Step 3: Set Marginal Profit to Zero

Since \( x_0 \) is the point of maximum profit, the derivative of the profit function at \( x_0 \) is zero: \( P'(x_0) = 0 \). This condition maximizes the profit.

Step 4: Equate Marginal Revenue and Marginal Cost

From the condition \( P'(x_0) = 0 \), we have \( R'(x_0) - C'(x_0) = 0 \). Solving this gives \( R'(x_0) = C'(x_0) \). This relationship shows that at the point of maximum profit, the marginal revenue equals the marginal cost.

Key Concepts

Understanding Marginal Revenue

Marginal revenue is vital in deciding how much of a product should be produced and sold. It explains how much additional revenue you would earn from selling just one more unit. Think of it as the increase in the total revenue as sales grow by one more unit. If you manufacture a new gadget and sell one extra piece, the extra money received is the marginal revenue.

This concept is fundamental, especially when the goal is profit maximization. By maximizing profit, businesses look for the optimal point where revenues from producing additional units start to level off or decrease. This happens when the marginal cost of producing one more unit equals the marginal revenue they gain from selling it.

• It helps in understanding the optimum production level.
• Guides companies on pricing strategies.
• Aids in evaluating the performance of marketing campaigns to boost sales.

Grasping what marginal revenue is and how it behaves makes it clear why businesses need to pay close attention to balance their costs and revenues properly.

Exploring Marginal Cost

Marginal cost reflects the cost of producing just one more unit of a product. It shows how much extra money you need to spend if you make an additional piece of your product. For example, if it costs you $100 to make 10 units but $105 to make 11, the marginal cost of the 11th unit is $5.

This understanding is essential to ensure that companies are not overspending to manufacture products that don't bring in enough revenue to cover costs. The idea is to find the best point where producing further doesn't cost more than what you gain from sales.

• Helps in deciding the scale of production efficiently.
• Ensures production does not exceed the capacity where costs outweigh the benefits.
• Assists in strategic planning for when to increase production based on demand.

Marginal cost connects directly with real-time business decisions to make sure each product sold is profitable.

Understanding Differentiable Functions

Differentiable functions play a major role in economics, particularly when working on revenue and cost functions. A function is differentiable if it has a derivative at each point in its domain, which means you can draw a smooth curve without any sharp corners.

In our context, understanding that revenue and cost functions are differentiable means that we can calculate their derivatives, which represent marginal revenue and marginal cost respectively. This smoothness in calculation allows businesses to analyze and find the optimum point of production where costs meet revenues perfectly.

• Ensures smooth analysis and decision-making.
• Makes it possible to find maximum or minimum points in business scenarios.
• Aids in calculating accurate rates of change like marginal revenue and cost.

Comprehending differentiable functions is essential, ensuring businesses can maximize profit effectively by letting them identify the exact point where costs and revenues are balanced correctly.