Questions: Determine the average cost function C̅(x) = C(x)/x. To find where the average cost is smallest, first calculate C̅'(x), the derivative of the average cost function. Then use a graphing calculator to find where the derivative is 0. Check your work by finding the minimum from the graph of the function C̅(x). C(x) = (1/2) x^3 + 3 x^2 - 3 x + 50 Determine the average cost function. C̅(x) =

Solution Steps

To minimize the average cost function, first express it as the total cost function divided by the quantity, then find its derivative. Use a graphing calculator to locate where this derivative equals zero, which indicates a minimum point, and confirm by observing the minimum on the graph of the average cost function.

The total cost function is given by

\[ C(x) = \frac{1}{2} x^3 + 3 x^2 - 3 x + 50. \]

The average cost function is defined as

\[ \overline{C}(x) = \frac{C(x)}{x} = \frac{\frac{1}{2} x^3 + 3 x^2 - 3 x + 50}{x} = \frac{1}{2} x^2 + 3 x - 3 + \frac{50}{x}. \]

The derivative of the average cost function is

\[ \overline{C}'(x) = \frac{1.5 x^2 + 6 x - 3}{x} - \frac{\frac{1}{2} x^3 + 3 x^2 - 3 x + 50}{x^2}. \]

Setting the derivative equal to zero, we find the critical points. The critical points are

\[ x \approx 2.9089, \quad x \approx -2.9545 - 2.9086i, \quad x \approx -2.9545 + 2.9086i. \]

Since we are interested in real values, we focus on

\[ x \approx 2.9089. \]

Final Answer