Part 1) Use mathematical induction to show that when n is an exact power of 2, the solution of the recurrence

??(??)={2 ,???? ??=22??(??2)+??,???? ??=2??,for ??>1 is T(n) = n lg n.

Part 2) Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following conjectures. To disprove give a counter example. a. If f1(n) = O(g(n)) and f2(n) = O(g(n)) then f1(n)= ? (f2(n) ). b. If f1(n) = O(g1(n)) and f2(n) = O(g2(n)) then f1(n)+ f2(n)= O(max{g1(n), g2(n)} )