A company determined that the marginal cost, $C^{'} (x)$ of producing the $x$ th unit of a product is given by $C^{'} (x) = x^{3} - 2 x$ . Find the total cost function $C$ , assuming that $C (x)$ is in dollars and that fixed costs are $$ 8000$ .
$C (x) =$

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Answer

Final Answer: $C (x) = \frac{x^{4}}{4} - x^{2} + 8000$

Steps

Step 1 ：The total cost function $C (x)$ is the integral of the marginal cost function $C^{'} (x)$ plus the fixed costs.

Step 2 ：In this case, we need to integrate $C^{'} (x) = x^{3} - 2 x$ and then add the fixed costs of $8000$ .

Step 3 ：The total cost function $C (x)$ is $x^{4} / 4 - x^{2} + 8000$ .

Step 4 ：Final Answer: $C (x) = \frac{x^{4}}{4} - x^{2} + 8000$