1. Prove that a function f is convex if and only if for all n 2 0, for all xo, 21, ..., In E dom(f), for all to, t1, . .., tn 2 0 such that ELoti = 1, n n f ( [ timi ) s [tif(Ii). i=0 i=0 2. Let f : Red -> R be a convex function and let g : R - R be a convex and nondecreasing function. Prove that the function go f : Rd -> R is convex. 3. Let f : R - R be a convex function and let o E R. Prove that the slope of the secant line between ro and x is a nondecreasing function of x. 4. Prove that x E R -> Ix|; ER is a convex function. 5. Let f : Rd -> R be a differentable convex function. Prove that if the gradient of f vanishes at some r*, then * is a minimizer of f