Determine the average cost function $$ \bar{C}(x)=\frac{C(x)}{x} $$. To find where the average cost is smallest, first calculate $$ \bar{C}^{\prime}(x) $$, 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 $$ \bar{C}(x) $$. $$ C(x)=\frac{1}{2} x^{3}+5 x^{2}-2 x+50 $$

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To find where the average cost is smallest, calculate the derivative of the average cost function $$\bar{C}(x)$$, set it to zero, and solve for $$x$$. Use a graphing calculator to find the roots of the resulting cubic equation and verify the minimum.

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