For each of the following utility functions:
a.) U (X,Y) = 3X + Y
b.) U (X,Y) = sqrt(XY)
c.) U (X,Y) = sqrt(X^2 + Y^2)
d.) U (X,Y) = ln(X) + ln(Y)
(i) Derive an equation for the marginal rate of substitution (MRS).
(ii) Calculate the MRS when the bundle is (X = 1 and Y =4) and the MRS when the bundle is (X = 4 andY = 1).
(iii) Based on your answers are preferences described by the utility function convex?