Problem 2 (40 points): (a) |25 points] Take the following functions and arrange them in ascending order of growth rates. That is, if function g(n) immediately follows function f(n) in your list, then it should be the case that f(n) = O(g(n)). Justify your answer. gl (n) 2 V logn Y2(n) = 2n 9s(n) = n4/3 a(n) = n(log n)3 9s(n) = nlogn 9s(n) = 22n gr(n) = 2n2 (b) [15 points] Assume that you have two functions f(n) and g(n) such that f(n) = 0(g(n)). For each of the following statements, decide whether you think it is true or false and accordingly give a proof (if true) or a counter-example (if false). (i) log2 f(n) is O(log2 g(n)), under the assumptions of f(n) 2 1 for all n, and log2 9(n) 2 1 for all n 2) is (2() (ii) f (n) is O (g(n)2), under the assumptions of f (n) 2 1 for all n, and log2 g(n) 2 1 for alln