Please see the attached problem

Consider the following problem minimize f (3:) subject to (I: E A\" (I) Where A\" C R" is the probability simplex, i.e., A\" = {:I: E R\" I 223:1 3:1,; : 1, :I: Z 0}, and f is a proper convex function such that relint(dom( f )) relint(An) % {3). Assume that f is differentiable over the relative interior of its domain. (a) Show that 33* E A\" is an optimal solution of the problem in (1) if and only if there exists A E R such that for all 1' : 1, . . . , n, aw) am) an 3331' 2A ifmg'zo :/\\ ifit': >0 and (b) Using part (a), determine a solution to the problem in (1) for the function f given by 23:1 5172' 111 331' (that) for :1: 2 0 f(:1:) : { +00 otherwise Where (I. E R\" is a given vector