Suppose that an optimal solution to the problem: minimize ex subject to ax > b, x > 0, exists. prove the following complementarity theorem. a. a variable is zero for all primal optimal solutions if and only if its complementary dual variable is positive for some dual optimal solution. b. a variable is unbounded in the primal feasible set if and only if its complementary dual variable is bounded in the dual feasible set. c. a variable is unbounded in the primal optimal set if and only if its complementary dual variable is zero for all dual feasible solutions. d. a variable is positive for some primal feasible solution if and only if its complementary dual variable is bounded in the dual optimal set.