For each of the following utility functions for goods x and y 3.1. calculate the marginal of utility for x; 3.2. calculate the marginal utility for y; 3.3. calculate the marginal rate of substitution; 3.4. graph two indifference curves; identify which one yields higher utility; 3.5. using the definition of convex preferences, verify whether the preferences are convex. Choose two bundles that belong to the same indifference curve. Label them a and b. Define a third bundle that mixes one half of each of the previous ones. Label it c. Evaluate whether it is at least as good as both a and b.

1. u(x, y) = y + x 2. u(x, y) = min{x, 2/3y} 3. u(x, y) = x^3/2, y^3/4