Let f and g be two functions that take integers as input and outputs real numbers. Big-Oh: f(n) is 0(g(n)) if and only if there exists a real constant c 0 and an integer no 0 such that f(n) cg(n) yn no- Big-Omega: f(n) is g(n)) if and only if there exists a real constant c 0 and an integer no > 0 such that f(n) 2 c.g(n) Vn 2 no- Big-Theta: f(n) is (g(n)) if and only if f(n) is 0(g(n)) and f(n) is (g(n)) Based on the definitions above, prove the followings. 1. 5n2 +6n + 12 is O(n3) 2. 5n2 6n 12 is 2(n2) 3, 5n2 + 6n + 12 is (n2) Prove the following theorems using the definition of Big-Oh from above l. R1 (Scaling): If f(n) is 0(g(n)) then af(n) is 0(g(n)), a 0. 2. R4 (Transitivity): If d(n) is O((n)) and f(n) is O(g(n)), then d(n) is Og(n) 3. R7: log(n*) is 0(log n) for any fixed x 0. 4. R6: nz is 0(an) for any fixed x 0 and a 1