   Chapter 2, Problem 34RE ### 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

# Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval. cos x = e x − 2 ,   ( 0 , 1 )

To determine

To show: There is a root of the equation cosx=ex2 in the interval (1,2).

Explanation

Theorem used: The Intermediate Value Theorem

Suppose that if 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.

Proof:

Rewrite the equation as, cosx=ex2.

To show there is a root of the equation cosx=ex2 in the interval (0,1), it is enough to show that there is a number c between 0 and 1 for which f(c)=0.

Consider the function f(x)=cosxex+2 and take a=0, b=1 and N=0.

Substitute 0 for x in f(x),

f(0)=cos0e0+2=11+2=2

This implies that, f(0)=2>0

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