   Chapter 4.2, Problem 29E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Show that sin x < x if 0 < x < 2π.

To determine

To show: The inequality of the function sinx<x if 0<x<2π .

Explanation

Mean value Theorem:

“If f be a function that satisfies the following hypothesis:

1. f is continuous on the closed interval [a,b] .

2. f is differentiable on the open interval (a,b) .

Then, there is a number c in (a,b) such that f(c)=f(b)f(a)ba .

Or, equivalently, f(b)f(a)=f(c)(ba) ”.

Proof:

Consider, the function f(x)=xsinx .

Apply the end points 0<x<2π in f(x) .

Substitute 0 for x in f(x) ,

f(0)=0sin(0)=0[sin(0)]=0

Substitute 2π for x in f(x) ,

f(2π)=2πsin(2π)=2π0[sin(nπ)=0]=2π

The derivative of f(x) is exist as f'(x)=1cosx

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