   Chapter 11.2, Problem 11ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# a. Show that for any integer n ≥ 1 ,   0 ≤ 23 n 4 + 8 n 2 + 4 n ≤ 35 n 4 . b. Show that for any integer n ≥ 1 ,   23 n 4 ≤ 23 n 4 + 8 n 2 + 4 n .c. Sketch a graph to illustrate the result of parts (a) and (b). d. Use the O- and Ω -notations to express the results of parts (a) and (b). e. What can you deduce about the order of 23 n 4 + 8 n 2 + 4 n ?

To determine

(a)

To prove:

The function 23n4+8n2+4n is greater than or equal 0 and less than or equal 35n4 for n1.

Explanation

Given information:

The functions 23n4+8n2+4n and 35n4 are defined on n and all n are positive integers where n1.

Proof:

Let f(n)=23n4+8n2+4n.

It is clear that for any integer n0, f(n) is greater than or equal zero.

As an example,

let n=1,f(n=1)=23(1)4+8(1)2+4(1)=23+8+4=35

Hence, f(1)0

Therefore, f(n)0 for any integer n0.

Also, when n=1, f(1)=35 as we proved previously.

Then,

f(n=1)=35=35(1)=35(1)4f(n=1)=35(n)4

Let’s assume that f(p)35p4 for any integer n=p where n>1.

Hence,

f(p)=23p4+8p2+4p35p4

when n=p+1 ,

f(p+1)=23(p+1)4+8(p+1)2+4(p+1)=23(p4+4p3+6p2+4p+1)+8(p2+2p+1)+4p+4=(23p4+8p2+4p)+(92p3+138p2+112p+35)f

To determine

(b)

To prove:

The function 23n4+8n2+4n is greater than or equal 23n4 for n1.

To determine

(c)

To graph:

An illustration about the variation of the functions in previous part(a.) and party(b.)

( 023n4+8n2+4n35n4 and 23n4+8n2+4n23n4 )

To determine

(d)

The relationship between the functions 23n4+8n2+4n, 35n4 and 23n4 in the notations of O and Ω.

To determine

(e)

The order of the functions 23n4+8n2+4n

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