An investor has wealth $ initially and is now facing a risky gamble. In this risky gamble, he can earn $ with probability and lose $ with probability 1, where >0 and 01. The investor can choose whether to enter into the gamble.

• If he chooses to gamble, he will get either $(+) or $().
• If he does not gamble, he will get $.

We let () (0) be the utility function of the investor. Here, () is a continuous function of . Intuitively, the gamble should be more attractive for the investor if the chance of winning is larger. Complete the following task by considering maximum expected utility criterion.

(a) If the investor is risk-neutral, find the range of such that the investor prefers gambling.

(b) If the investor is risk-averse, show that there exists 0(12,1) such that the investor strictly prefers gambling if and only if >0.

(c) If the investor is risk-lover, show that there exists 0(0,12) such that the investor strictly prefers gambling if and only if >0.