To solve this, we need to follow a few steps to find the Marginal Revenue (MR) function and the profit-maximizing quantity for Gamerix. Let's break it down.

Step 1: Demand function

The demand function is given as: $P = 10 - Q$ This represents the relationship between price $P$ and quantity $Q$.

Step 2: Total Revenue (TR)

The total revenue is given by the product of price $P$ and quantity $Q$: $TR = P \times Q = (10 - Q) \times Q$ $TR = 10Q - Q^2$ This is the total revenue function.

Step 3: Marginal Revenue (MR)

Marginal Revenue is the derivative of Total Revenue with respect to $Q$: $MR = \frac{d(TR)}{dQ}$ Differentiate the total revenue function: $MR = \frac{d}{dQ}(10Q - Q^2) = 10 - 2Q$

Step 4: Marginal Cost (MC)

The total cost function is given as $C(Q) = 4Q$. The Marginal Cost (MC) is the derivative of the total cost with respect to $Q$: $MC = \frac{dC(Q)}{dQ} = 4$

Step 5: Profit Maximization

A profit-maximizing monopolist produces where Marginal Revenue equals Marginal Cost (MR = MC): $10 - 2Q = 4$ Solve for $Q$: $10 - 4 = 2Q$ $Q = 3$

Final Answer:

• Marginal Revenue function: $MR = 10 - 2Q$
• Profit-maximizing quantity: $Q = 3$

Would you like a detailed explanation of how we derived the profit-maximizing condition?

---

Here are 5 related questions:

1. How do we calculate the price Gamerix charges at the profit-maximizing quantity?
2. What is the total revenue Gamerix earns at the profit-maximizing quantity?
3. How does Gamerix's cost function influence its pricing strategy?
4. What is the relationship between marginal cost and total cost?
5. How would a shift in demand affect the profit-maximizing quantity?

Tip: In monopoly pricing, the marginal revenue curve always lies below the demand curve because the monopolist must lower the price to sell more units.