To solve this problem:

We are given the marginal cost function $MC(q) = 5q^4 + 12q + 10$ and the fixed cost is $600$. The total cost function $C(q)$ can be found by integrating the marginal cost function and then adding the fixed cost.

Step 1: Integrate $MC(q)$

The total cost function is the integral of the marginal cost function: $C(q) = \int MC(q) \, dq + \text{Fixed Cost}$

So, $C(q) = \int \left( 5q^4 + 12q + 10 \right) dq + 600$

Step 2: Compute the integral term by term

$\int 5q^4 \, dq = \frac{5}{5}q^5 = q^5$ $\int 12q \, dq = \frac{12}{2}q^2 = 6q^2$ $\int 10 \, dq = 10q$

Thus, $\int MC(q) \, dq = q^5 + 6q^2 + 10q$

Step 3: Add the fixed cost

Now include the fixed cost of 600: $C(q) = q^5 + 6q^2 + 10q + 600$

Final Answer:

$C(q) = q^5 + 6q^2 + 10q + 600$

Let me know if you'd like a detailed breakdown of any of the steps.

Related Questions:

1. How do you calculate fixed costs from a total cost function?
2. What is the economic interpretation of marginal cost?
3. How do you verify integration results for accuracy?
4. What happens to total costs if marginal cost is constant?
5. Can marginal cost functions involve trigonometric terms?

Tip:

Always double-check the constants when solving integration problems involving fixed costs, as they significantly affect the final total cost function.