   Chapter 4.2, Problem 9E ### 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

# Let f(x) = 1 − x2/3. Show that f(−l) = f(1) but there is no number c in (−1, 1) such that f′(c) = 0. Why does this not contradict Rolle’s Theorem?

To determine

To show: For the function f(x)=1x23 , f(1)=f(1) but it has no number c in the open interval (1,1) such that f(c)=0 ; justify that why this not contradict the Rolle’s Theorem.

Explanation

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 .”

Given:

The function f(x)=1x23 .

Proof:

Find the values of f(x) for the end points of the given interval.

Substituting x=1 in f(x) ,

f(1)=1(1)23=11=0

Substituting x=1 in f(x) ,

f(1)=1(1)23=11=0

Hence, it is proved that f(1)=f(1)

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