   Chapter 2.5, Problem 53E ### 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.x4 + x – 3 = 0, (1,2)

To determine

To show: There is a root of the equation x4+x3=0 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:

In order to show that there is a root of the equation x4+x3=0 in the interval (1,2), it is enough to show that there is a number c between 1 and 2 for which f(c)=0.

Consider the polynomial f(x)=x4+x3 and take a=1, b=2 and N=0.

Substitute 1 for x in f(x),

f(1)=(1)4+(1)3=1+13=23=1

This implies that, f(1)=1<0

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