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

Answer

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Answer

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

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 $C'(x) = x^3 - 2x$ 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: $C(x) = \boxed{\frac{x^4}{4} - x^2 + 8000}$