   Chapter 4.2, Problem 19E ### 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 the equation has exactly one real root.2x + cos x = 0

To determine

To show: The equation 2x+cosx=0 has exactly one root.

Explanation

Theorem used:

Intermediate Value Theorem:

“Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b) , where f(a)f(b) . Then, there exists a number c in (a,b) such that f(c)=N ”.

Rolle’s Theorem:

“Let a function f satisfies the following conditions,

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

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

3. f(a)=f(b)

Then, there is a number c in open interval (a,b) such that f(c)=0 .”

Proof:

The given equation is 2x+cosx=0 .

Rearrange the terms of the equation as, cosx=2x .

Solve for x.

Recall the fact that, cosx always lies in the closed interval [1,1] .

That is, 1cosx1 .

Since cosx=2x , the above inequality becomes 12x1 .

Dividing the inequality by 2 yields, 122x212

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 