Which of the utility functions in Exercise 7 are concave and which are convex? (DeGroot XXXXXXXXXXAssume that Mr. A and Mr. B have the same utility function for a change, x, in their fortune, given by...

Which of the utility functions in Exercise 7 are concave and which are convex?

(DeGroot (1970)) Assume that Mr. A and Mr. B have the same utility function for a change, x, in their fortune, given by
  Suppose now that one of the two men receives, as a gift, a lottery ticket which yields either a reward of r dollars (r > 0) or a reward of 0 dollars, with probability
 each. Show that there exists a number b > 0 having the following property: Regardless of which man receives the lottery ticket, he can sell it to the other man for b dollars and the sale will be advantageous to both men.