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Show that (x2 + xy + x log y)3 is O(x6y3).

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 53E Chapter 3.2

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 53E

Show that (x2 + xy + x log y)3 is O(x6y3).

Step-by-Step Solution:
Step 1 of 3

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 .

Step 2 of 3

Chapter 3.2, Problem 53E is Solved
Step 3 of 3

Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Show that (x2 + xy + x log y)3 is O(x6y3).