Suppose that \( c(x)=3 x^{3}-18 x^{2}+24,000 x \) is the cost of manufacturing \( x \) items. Find a production level that will minimize the average cost of making \( x \) items.

To find the production level that minimizes the average cost of making \( x \) items, we first need to determine the average cost function, \( A(x) \). The average cost is given by the total cost divided by the number of items produced: \[ A(x) = \frac{c(x)}{x} = \frac{3x^3 - 18x^2 + 24000x}{x} \] For \( x > 0 \), we can simplify this expression: \[ A(x) = 3x^2 - 18x + 24000 \] Next, we need to find the critical points of \( A(x) \) by taking its derivative and setting it to zero. The derivative \( A'(x) \) is: \[ A'(x) = \frac{d}{dx}(3x^2 - 18x + 24000) = 6x - 18 \] Setting the derivative equal to zero to find critical points: \[ 6x - 18 = 0 \] Solving for \( x \): \[ 6x = 18 \\ x = 3 \] To determine whether this critical point is a minimum, we can check the second derivative \( A''(x) \): \[ A''(x) = \frac{d^2}{dx^2}(3x^2 - 18x + 24000) = 6 \] Since \( A''(x) = 6 > 0 \), the function \( A(x) \) is concave up at \( x = 3 \), indicating that this point is indeed a minimum. Thus, the production level that minimizes the average cost of making \( x \) items is: \[ \boxed{3} \]