Show that the function f .x/ D 2x is convex. Consider RANDOMIZED-QUICKSORT operating on a sequence..

Show that the function f .x/ D 2x is convex. Consider RANDOMIZED-QUICKSORT operating on a sequence of n distinct input numbers. Prove that for any constant k>0, all but O.1=nk/ of the nŠ input permutations yield an O.n lg n/ running time.