Exercise 3Consider a monopolist firm that faces an inverse demand P = Q/2 + 5, and has a total cost function C(Q) = 2/3Q³ 2Q² + 6Q. The domain of the cost function, profit function, and revenue function is [0, 10].

1. Find the monopolist's profit function, (Q).

2. Find and sketch the marginal revenue function MR(Q), the marginal cost function MC(Q), and the marginal profit function (Q) in one graph. If there are intersections with the vertical axis and with the horizontal axis, please highlight them clearly.

3. In your graph, highlight the part of the domain where the profit function (Q) is convex and that where (Q) is concave. Moreover, highlight the part of the domain where the cost function C(Q) is convex and that where C(Q) is concave. Also highlight the part of the domain where the revenue function R(Q) is convex and that where R(Q) is concave.

An unexpected pandemic changes the market demand for the monopolist's product and also forced the monopolist to change its production technology. Due to these changes, the profit function of the firm becomes (Q) = 2Q² + 8Q, with the domain of the profit function still being [0, 10].

4. Using the first-order condition to find the critical points, i.e., all candidates for local extrema of the new profit function.

5. Are the critical points you found above local maximizers? Explain your answer. As the status of the pandemic improves, the market condition faced by the monopolist improves, and the profit function of the firm changes to ( Q), which is a concave function on the domain [0, 10]. Suppose that the function is increasing at 10, i.e., (10) > 0.

6. How many critical points does ( Q) have? Explain your answer.

7. Find the profit-maximizing quantity Q for the profit function ( Q).