Math Snap

STEP 1

1. The marginal cost function is given by $C^{\prime}(x) = x^3 - 4x$.
2. The fixed cost is $8000.
3. We need to find the total cost function $C(x)$.

STEP 2

1. Integrate the marginal cost function to find the total cost function $C(x)$.
2. Determine the constant of integration using the fixed cost information.
3. Write the complete expression for the total cost function $C(x)$.

STEP 3

Integrate the marginal cost function $C^{\prime}(x) = x^3 - 4x$ to find $C(x)$:
$$C(x) = \int (x^3 - 4x) \, dx$$

STEP 4

Perform the integration:
$$C(x) = \int x^3 \, dx - \int 4x \, dx$$ $$C(x) = \frac{x^4}{4} - 2x^2 + C_0$$ where $C_0$ is the constant of integration.

STEP 5

Use the fixed cost information to determine the constant of integration. Given that the fixed cost is $8000, we have:
$$C(0) = 8000$$ Substitute $x = 0$ into the total cost function:
$$C(0) = \frac{0^4}{4} - 2(0)^2 + C_0 = 8000$$ $$C_0 = 8000$$

Substitute $C_0 = 8000$ back into the expression for $C(x)$:
$$C(x) = \frac{x^4}{4} - 2x^2 + 8000$$ The total cost function $C(x)$ is:
$$C(x) = \frac{x^4}{4} - 2x^2 + 8000$$