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. \(C=2 x^{2}+5 x+18\)

Short answer

The production level at which the average cost is minimum is when \(x = 3\). At this production level, the marginal cost equals the average cost, which is 17 units.

Step-by-step solution

Derive the Average Cost Function

The average cost (AC) function is defined as the total cost (C) divided by the quantity (x). In this case, the total cost function is \(C=2 x^{2}+5 x+18\). Therefore, \[ AC=\frac{C}{x} = \frac{2 x^{2}+5 x+18}{x} = 2x + 5 + \frac{18}{x} \]. This is our average cost function.

Find the Minimum of the Average Cost Function

To find the minimum of \(AC\), we take the derivative \(AC'\) and set it equal to zero.So, \(AC'=2 - \frac{18}{x^2}\). Setting this equal to zero and solving for \(x\) gives \(x = \sqrt{9}\) or \(x=3\). So, when the quantity \(x=3\) items, the average cost is at its minimum.

Derive the Marginal Cost Function

The marginal cost (MC) is the derivative of the total cost function \(C\). So, \(MC = C' = 4x + 5\). This is our marginal cost function.

Equate Marginal Cost to Average Cost

At the level of production where average cost is minimum, the marginal cost equals the average cost. So we set Marginal Cost (MC) equal to Average Cost (AC) when \(x = 3\) to verify from the previous step. Plugging \(x=3\) into \(MC\) gives \(MC = 4(3) + 5 = 17\), and plugging into \(AC\) gives \(AC = 2(3) + 5 + \frac{18}{3} = 17\). Since \(MC = AC = 17\), this confirms that the marginal cost equals the average cost at the production level that minimizes the average cost.

Graph the Average Cost Function

Using a graphing utility, graph the average cost function, \(AC = 2x + 5 + \frac{18}{x}\). The minimum point on the curve verifies our result that the minimum average cost occurs at \(x = 3\). Moreover, when Average Cost is plotted against quantity, the Marginal Cost curve intersects exactly at the level of output where Average Cost is at its minimum i.e. at \(x = 3\).