I need help with this question, and please don't forget to make the comment requested on part c. Please make sure to answer parts c, d and e. Thank you

David and Julie are two close friends who like collecting bicycles and balloons. David's preferences for bicycles (good x) and balloons (good y) are represented by the utility function u(x, y) = xy². Suppose that the unit prices p_x and p_yare 1. Their income to spend is 6 million dollars for each.

(a) Solve for David's optimal choice.

(b) Suppose p_xrises to 2 dollars. What is his new optimal consumption plan? How much would he have to increase his income in order to remain indifferent to the original situation?

(c) Julie's preferences are represented by v(x, y) = ln x + 2 ln y1000000. Answer parts (a) and (b) for Julie and comment.

(d) In fact, generalizing the conclusion from part (c), suppose that a consumer's utility function is u, and that f is an order-preserving transformation of u. Using the chain rule of differentiation, can you prove that, at any bundle, the MRS of the transformed utility function f(u) is the same as that of the original function u?

(e) Finally, using the Lagrange function method, what is David's marginal utility of income at the optimum,, in the initial situation? What about in thefinal situation?