Show that (x2 + xy + x log y)3 is O(x6y3).
Solution:Step-1: In this problem we need to show that is Note: Let us consider f and g are functions from the set of integers to the set of real numbers. The estimate value can be said that f(x) is O(g(x)) if there are constants C and k such that , where C > 0 and x> k.The constants C and k are called the witnesses to the relationship. The definition of f(x) is O(g(x)) says that f(x) grows slower than some fixed multiple of g(x) as x grows without bound.Step-2:Consider , ………….(1)Assume that x > 1.We know that for all x >1 .…………(2)For every y > 0 , log(y) < y ………..(3)From (1), (2) , and (3) we get: , forall x > 1and y > 1 . Therefore , Step-3: By using condition we get: Cubing on both sides we get: Therefore , .So, by using the above note it is clear that C = 27 and .Therefore .