Definition 1.A functionf:XR on a convex setXRn is said to be quasiconcave if for each cR, the following set is convex: {x|f(x)c}.

We didn't talk about the following concept at all, although it's closely related:

Definition 2.A function f:XR on a convex set XRn is said to be quasiconvex if for each cR, the following set is convex: {x|f(x)c}.

(a) Are quasiconvexity and quasiconcavity equivalent: i.e., is a function quasiconcave if and only if it is quasiconvex? Justify your answer or provide a counter example.

(b) Are there functions which are both quasiconvex and quasiconcave? Justify your answer.

(i) A good answer to this question provides an example.

(ii) A great answer to this question provides a general characterization of the set functions which areboth quasiconcave and quasiconvex.

(c) Are all convex functions quasiconvex? Hint: we discussed the relationship between concavity andquasiconcavity in class.

(d) Are all quasiconvex functions convex? Hint: see the hint in part (c).