For each of the following utility functions, calculate (or figure out) the MRS and sketch out an indifference curve for U=4, plotting at least 2 points on each curve, and explain whether the preferences violate the diminishing MRS or convexity of indifference curves. Also make clear if the indifference curve hits the x and/or y axis.
a) U(x, y) = 2ln x + (ln y)/2
b) U (x ,y) =2 x+(1/2)y
c) U(x, y) = min(2x, y/2)
d) U(x, y) =2sqrt( x) + y/2