Question

Use the cost function to find the production level for which the average cost is a minimum. For this production level, show that the marginal cost and average cost are equal. Use a graphing utility to graph the average cost function and verify your results. \(C=x^{3}-6 x^{2}+13 x\)

Short answer

The production level for which the average cost is a minimum is 3. At this production level, the marginal cost and average cost are both equal to 3. Graphing the AC function on a graphing utility confirms these results.

Step-by-step solution

Derive the Average Cost (AC) function

The average cost function AC is the cost function C divided by the quantity x. So, \(AC=C/x=( x^{3}-6x^{2}+13x)/x = x^{2}-6x+13\)

Derive the Marginal Cost (MC) function

This is the derivative of the cost function. So, \(MC = dC/dx=3x^{2}-12x+13\)

Derive the Production Level for Minimum AC

Find the derivative of the AC function, set it equal to zero and solve for x to find the production level for minimum AC. The derivative, \(dAC/dx = 2x-6\). Setting to zero gives\(2x-6=0\),which gives \(x = 3\). Thus, the production level for minimum AC is 3.

Verify Marginal Cost Equals Average Cost

Substitute the value of x (found in step 3) into the MC function to determine if it yields the same result as the AC. \(MC = 3*(3)^{2}-12*3+13 = 3\). Substituting \(x=3\) in the AC equation also yields 3. Thus, as required, Marginal Cost equals Average Cost.

Graph the AC function

Using any graphing utility, graph the function \(AC = x^{2}-6x+13\). The minimum point of the curve will correspond to \(x=3\), which verifies the previously computed result.

Key concepts

Cost Function

The cost function is a mathematical expression used to calculate the total cost of producing a certain number of goods, or units, in a business. It allows a company to understand how different levels of production impact costs, helping to guide budgeting and pricing decisions. In this context, the cost function is given by \[ C = x^3 - 6x^2 + 13x \]where \( C \) is the total cost and \( x \) represents the quantity of the product produced.

• This polynomial expression highlights that costs are not always linear. Various levels of production affect cost differently due to the different powers of \( x \).
• Understanding the cost function helps in identifying the relationship between production quantities and cost. As such, businesses can better strategize their production to maintain profitability.

To gain deeper insights, it's essential to break down components of this function, especially when aiming to identify the production level with the minimum average cost.

Marginal Cost

Marginal cost refers to the additional cost incurred to produce one more unit of a product. It's a crucial part of cost analysis in economics because it reflects how variable costs behave as production volume changes. Here, the marginal cost is determined by differentiating the cost function with respect to \( x \):\[ MC = \frac{dC}{dx} = 3x^2 - 12x + 13 \]

• This derivative tells us the rate at which costs are changing as production levels increase.
• A key insight is that when marginal cost equals the average cost, it marks the optimum production level for minimizing average costs.

By analyzing the marginal cost, businesses can make informed decisions regarding production adjustments, especially when planning to scale operations.

Production Level

The production level that minimizes the average cost is pivotal as it guides businesses to operate efficiently. From the derived average cost function:\[ AC = \frac{C}{x} = x^2 - 6x + 13 \]we find the production level that minimizes this average cost by setting the derivative of the average cost function to zero:\[ \frac{dAC}{dx} = 2x - 6 = 0 \]Solving gives \( x = 3 \).

• This indicates that at the production level of \( x = 3 \), the average cost is minimized.
• At this point, both marginal cost and average cost equal each other, confirming the point of optimal efficiency.

Understanding this level is key in optimizing resource allocation and managing operational efficiency in production.

Graphing Utility

Graphing utilities are powerful tools for visualizing mathematical functions, providing clear visual proof of theoretical conclusions. In this exercise, graphing the average cost function:\[ AC = x^2 - 6x + 13 \]can effectively demonstrate the production level where the average cost is minimized.

• By using a graphing calculator or software, we can visually verify that the minimum point of the curve occurs at \( x = 3 \).
• This visual evidence reinforces analytical solutions and helps gain a better grasp of abstract mathematical concepts.

Using a graphing utility not only simplifies complex calculations but also aids in learning by providing an intuitive understanding of cost behavior across different production levels.