For (1) there is actually nothing to prove in my opinion. If maximizes f(x) s.t. h(x) = c then it is clear that (II) follows from (I)

For (2) imagine a function that has its absolute maximum in the negative domain of and decreases the closer it gets to zero and even decreases further for any positive number. Let this e.g. be for : . If you maximize this function for x you will get . However, if you can only choose from the positive numbers (you might want to use Kuhn Tucker to prove this formally) your . However, this differs to the solution for (I) thus you proved that (2) is wrong by counter example.