Do you think diminishing marginal utility is a necessary

Question

Find the  MU_x , MU_y, and MRS equations for each of the following utility functions.

a. U = x^0.6y^0.4.

b. U = x² + y²x,y>0.

c. U = 2x + 4y.

d. U = x²y².

e. U = x^ay^b.

1. For (a)-(d), which of the  utility functions exhibit diminishing marginal utility for good X? Hint: Using your equation for MU_x, determine if MU_x falls as X rises.

2. Which of the above utility functions exhibit diminishing MRS? (That is, which of the above yield convex indifference curves?).

3. Do you think diminishing marginal utility is a necessary condition to get diminishing MRS? Use your answers for (a), (d), and (e) to justify your answer.