5. (24 marks) Find the first and second derivatives of each of these functions. Simplify as much as possible. a) (8 marks) f(z) = In(z + %) b) (8 marks) g(z) = (223 z)e ) (8 marks) h(z) = (3*)~1 6. (18 marks) Use implicit differentiation to find g% and %'21 for each of the following cases (the second derivative should be in terms of z and y, not ): a) (8 marks) 4y% +y = 2z b) (10 marks) e +y = % + z? 7. (12 marks) Suppose that a firm can sell a quantity at a price of 5000 30q. The firm's costs are given by C(q) = 5 + 2% + F for some F > 0. The constant F is a fixed cost that only needs to be paid if the firm opens its doors (think of this as registering the firm, buying land for production, etc.). If the firm does not open its doors, it earns 0 profits. a) (6 marks) Suppose that the firm has chosen to open its doors. What is the level of that maximises profits? b) (6 marks) For what values of F' should the firm open its doors? In other words, for what values of F' will the firm earn positive profits given the solution to part a)