For each of the following, answer true or false. If true, prove. If false, derive a counterexample

1. Consider the maximization of f(x) for some f C 1 over the domain D. If x is the solution and x int(D) then f (x ) = 0 is not a necessary condition.

2. Consider some function f. If f is differentiable at x then f is continuous at x.

3. If a function f is homogeneous of degree k > 0, then f is homogeneous of degree k 1. 1

4. If f(x) is strictly convex then f(x + (1 )y) max{f(x), f(y)} for all x y and for all (0, 1).

5. If A is convex and B is convex then A B is convex.

6. Let X1, X2 and X3 form a partition for X and let Y1, Y2 and Y3 form a partition for Y . If Xi Yi for all i, then X Y .

7. If f is a linear function then f is both weakly concave and weakly convex.