1. (20 points) Given the function f(x) = ln(x + 1),x > 0 a) Find the first derivative. Based on your answer, discuss the function in terms of being increasing/decreasing, its convex or concave properties. b) Find the second derivative. Based on your answer, discuss if the function is convex or concave. c) (Bonus: 2 points) In applying mathematical methods to economics/business problems, we use a broad menu of functions to accurately specify the properties of the economic variables in question. Provide an example that demonstrates the use of the given function to describe a certain economic or business activity. 2. (20 points) Elon Musk asks you to invest $10M in his new cyber scooter product. He promises that he will return $15M to you at the end of the next 5 years. On the other hand, your good friend and hedge fund manager Jim Simons wants you to invest in his new fixed income fund. Jim promises that your money with $10M will be continuously compounded over the next 5 years. At what interest rate would Jim's offer be better than that of Elon's? 3. (20 points) The market for a particular good can be described by the equations: Qa = 20-4P Qs 2+P w Qs = Qa = Q Where Q and P are respectively the equilibrium quantity and price, w is the current wage of labor used for production. a) Use repeated substitution to solve for the equilibrium Q and P as a function of w. b) Based on a), find the first derivative of function P(w). c) (Bonus: 3 points) Discuss the impact of a $1 increase in wage on price P. 4. (20 points) Babydrink is a monopolist due to its patent in infant formula. The total cost for production in dollars is given by C(q) = 24q + 600q + 9600. a) What is the fixed cost? b) Find a function that gives the marginal cost. c) Find a function that gives the average cost. d) Find the efficient scale (minimizes the average cost). e) Verify it is the most efficient, not the least efficient. 5. (20 points) Babydrink is a monopolist due to its patent in infant formula. The total cost for production in dollars is given by C(q) = 24q + 600q + 9600. The price-demand function is p = 700-q. (a) To maximize profit, how much should Babydrink produce? And what is the price it charges? (b) Verify that the answer from part (a) provides a maximum and not a minimum.