Microeconomic Theory 11th Edition by Walter Nicholson,Christopher Snyder

Exercise 2

In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing MRS (i.e., to be strictly quasi-concave), the following condition must hold:

Use this condition to check the convexity of the indifference curves for each of the utility functions in Problem 3.1. Describe the precise relationship between diminishing marginal utility and quasi-concavity for each case.
Reference: Problem 3.1
Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the MRS declines as x increases).