Can you please solve this, with the step by step answers! Thank you so much!

5. Assume a consumer's preferences are given by the following utility functionU(X,Y) = X²Y

A) Find the marginal rate of substitution between good X and good Y [MRS(X,Y)].Show all work

MRS(X,Y) = MUx/MUy = (U/X)/(U/Y) => MRS(.) = 2*(Y/X)

B) Let P_xbe the price of good X, P_ybe the price of good Y, and M be income. Find the individualconsumer demand functionsfor good X and good Y, and the equilibrium level of utility.Show all work

2 Conditions for utility maximization: 1) P_xX + P_yY = M and 2) MRS(.) = 2*(Y/X) = P_x/ P_ysolving1) and 2) simultaneously yields the following demand functions for good X and good Y

X* = 2/3(M/P_x) and Y* = 1/3(M/P_y)

C) If P_x=$2 P_y=$1 and M=$60 how many units of good X and good Y will maximize thisconsumers utility S.T. thebudget constraint? What is the consumer's equilibrium level of utility (i.e. U*(X*,Y*))? What is the relative price of good X in terms of good Y?

X₁* = 20 and Y₁* = 20.U₁*(20,20) = 20²*20 =8000.

The relative price of good X in terms of good Y is 2 units of good Y.

D) Show your solution to part C) graphically (be careful).

E) Assume now the price of good Y rises to $2 and nothing else changes, redo parts C) - D).

(use the same graph).

X₂* = 20 and Y₂* = 10.U₁*(20,20) = 20²*10 =4000.

The relative price of good X in terms of good Y is 1 unit of good Y.