   Chapter 11.1, Problem 21ES ### 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
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# a. Let m be any positive integer, and define f ( x ) = x m for each nonnegative real number x. Use the binomial theorem to show that f is an increasing function. b. Let in and ii be any positive integers, and let g ( x ) = x m / n for each nonnegative real number x. Prove that g is an increasing function. Note: The results of exercise 21 are used in the exercises for Sections 11.2 and 11.4.

To determine

(a)

To show:

Use the binomial theorem to show that f(x)=xm for all non-negative real numbers x .Use the binomial theorem to show that f is an increasing function.

Explanation

Given information:

f(x)=xm for all non-negative real numbers x

Concept used:

If u<vf(u)<f(v),f(x) is an increasing function.

Calculation:

Suppose u and v are non-negative real numbers with u<v.

To prove that f(x) is increasing, we must show that f(x)<f(v).

Let v=u+h for some positive real number h

By substituting in xm ,

Vm=(u+h)m=um+[( m 1 )um1.h+( m 2 )Um2h2+

To determine

(b)

To show:

Prove that g is an increasing funciton.

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