   Chapter 2.5, Problem 55E ### 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 given equation in the specified interval.ex = 3 – 2x, (0,1)

To determine

To show: There is a root of the equation ex=32x in the interval (0,1).

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, ex+2x3=0.

To show there is a root of the equation ex+2x3=0 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)=ex+2x3 and take a=0, b=1 and N=0.

Substitute 0 for x in f(x),

f(0)=e(0)+2(0)3=13=2

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

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