Let A R^n be a convex set. We have learned the following definition for quasi-concave functions:

Definition 1. f : A R is a quasi-concave function if and only if for any c R, the set {xA| f(x)c} is convex.

Here is an alternative definition:

Definition2. f:AR is a quasi-concave function if and only if for any x1,x2A and any

(0,1), f (x1 +(1)x2) min{f (x1), f (x2)}. Please prove that these two definitions are equivalent.

Hint: To show equivalence, you need to prove two parts: (1) if f satisfies the condition in Definition 1, then it must satisfy the condition in Definition 2; (2) if f satisfies the condition in Definition 2, then it must satisfy the condition in Definition 1.