Questions: A. 7 items B. 8 items C. 9 items D. 10 items Suppose c(x) = x^3 - 16x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost per item of making x items.

Solution Steps

The cost function is given by

\[ c(x) = x^3 - 16x^2 + 20000x. \]

The average cost function \( A(x) \) is defined as the total cost divided by the number of items produced:

\[ A(x) = \frac{c(x)}{x} = \frac{x^3 - 16x^2 + 20000x}{x} = x^2 - 16x + 20000. \]

To find the production level that minimizes the average cost, we first compute the derivative of the average cost function:

\[ A'(x) = 2x - 16. \]

Setting the derivative equal to zero to find critical points:

\[ 2x - 16 = 0 \implies x = 8. \]

Next, we evaluate the second derivative to confirm that this critical point is a minimum:

\[ A''(x) = 2. \]

Since \( A''(8) = 2 > 0 \), this indicates that \( x = 8 \) is indeed a minimum.

Final Answer