Q1. (True/False and Why, 25pts) For each of the following claims, state whether it is true or false (2pts) provide a short proof or counterexample justifying your claim (3pts). An insightful justification for a wrong true/false answer can still get you the latter 3pts. (a) Every global minimizer r* e Rd of a function f: Rd - R has Vf(x*) = 0. (b) For any function f : R - R and point r e Ro, prox (@) is a singleton. (c) For any twice continuously differentiable function f: Rd - R, if r* e Rd has V?f (x*) > 0 and Vf(x*) =0, then * is a local minimizer of f. (d) Every convex function f: R - R that is bounded below has a global minimizer. (e) For any convex f: R" - R and point a e Rd, if some point * ( Rd has p(x - I* ) e of (*) then * globally minimizes f(x) + 9//x - |3