1. Suppose that a real-valued function f on an open interval J in R has a second derivative f II on J . Show that f is convex if and only if f II ≥ 0 everywhere on J .

2. Let f (x, y) := x 2 y2 for all x and y. Show that although f is convex in x for each y, and in y for each x , it is not convex on R2.

3. If f and g are two convex functions on the same domain, show that f + g and max( f, g) are convex. Give an example to show that min( f, g) need not be.