   Chapter 4.2, Problem 21E ### 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 x3 − 15x + c = 0 has at most one root in the interval [−2, 2].

To determine

To show: The given equation has at most one root in the interval [2,2] .

Explanation

Given:

The equation is x315x+c=0 .

Theorem used: Rolle’s Theorem

“If 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:

Suppose, f(x) has two real distinct roots a and b where a<b on the interval [2,2] . Then f(a)=f(b)=0 .

Since the polynomial is continuous on the closed interval [a,b] and differentiable on open interval (a,b) , then by Roll’s Theorem, there exist a number c in (a,b) such that f(c)=0 .

Obtain the derivative of f(x)

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