Let’s analyze this problem step by step.

Key Information:

• Maximum profit is achieved at $q = 8$.
• Marginal revenue $MR(q) = 3$ at this profit-maximizing production level.
• We need to determine which of the given cost or marginal cost functions could be true by comparing their properties and constraints.

Approach:

Profit maximization occurs when: $MR(q) = MC(q)$ At $q = 8$, $MR(q) = 3$. Therefore, the marginal cost at $q = 8$ must also be $MC(8) = 3$. We evaluate the options using this principle.

Evaluating Each Option:

1. Option (i): $C(q) = \frac{q^2}{2} - 5q$, maximum production is 12 units

   • The marginal cost $MC(q)$ is derived as: $MC(q) = \frac{d}{dq} \left( \frac{q^2}{2} - 5q \right) = q - 5$
   • At $q = 8$: $MC(8) = 8 - 5 = 3$
   • This satisfies $MR(8) = MC(8) = 3$. Hence, this option could be true.

2. Option (ii): $C(q) = 2q$, maximum production is 8 units

   • The marginal cost $MC(q)$ is derived as: $MC(q) = \frac{d}{dq} (2q) = 2$
   • At $q = 8$: $MC(8) = 2$
   • Since $MR(8) = 3$ and $MC(8) \neq 3$, this option is not true.

3. Option (iii): $MC(q) = 4$, maximum production is 8 units

   • Here, the marginal cost $MC(q)$ is constant at 4 for all $q$.
   • At $q = 8$: $MC(8) = 4$
   • Since $MR(8) = 3$ and $MC(8) \neq 3$, this option is not true.

4. Option (iv): $MC(q) = \sqrt{q} + 1$, maximum production is 15 units

   • At $q = 8$, the marginal cost is: $MC(8) = \sqrt{8} + 1 = 2.83 + 1 \approx 3.83$
   • Since $MC(8) \neq 3$, this option is not true.

5. Option (v): None of these

   • Since option (i) satisfies the conditions, this option is not correct.

Correct Answer:

The only correct option is (i).

Let me know if you'd like further clarification!
Here are 5 follow-up questions to deepen your understanding:

1. How is the marginal cost derived from the cost function?
2. Why is the condition $MR = MC$ critical for profit maximization?
3. How can you verify if a production level truly maximizes profit?
4. What happens if $MR > MC$ or $MR < MC$?
5. What are the differences between marginal cost and marginal revenue?

Tip: Always compare $MR$ and $MC$ at the critical point to confirm if profit maximization occurs at the given production level.