Question

Production Level Suppose \(c(x)=x^{3}-20 x^{2}+20,000 x\) is the cost of manufacturing \(x\) items. Find a production level that will minimize the average cost of making \(x\) items.

Short answer

The production level that will minimize the average cost of making \(x\) items is \(x = 10\).

Step-by-step solution

Compute the average cost function

The average cost function can be obtained by dividing the given cost function by \(x\). Hence, the average cost function \(A(x) = c(x)/x = (x^{3}-20 x^{2}+20,000 x)/x = x^{2}-20 x+20,000\.

Find the derivative of the average cost function

The derivative of the average cost is obtained using the power rule and the constant rule. Hence, \(A'(x) = 2x - 20\).

Find critical points

To find the critical points, set \(A'(x) = 0\) and solve for \(x\) to find values of \(x\) that yield a horizontal tangent line. Solving \(2x-20 = 0\) gives \(x = 10\).

Confirm the minimum point

Derive second version A''(x) = 2 which is always positive. So, x = 10 is indeed at a minimum point.