   Chapter 3.2, Problem 9E

Chapter
Section
Textbook Problem

# Let f ( x ) = 1   − x 2 / 3 . Show that f ( − 1 ) = 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:

f-1=f(1) but there is no number c in (-1, 1) such that f'c=0 and explain why this does not contradict the Rolle’s theorem

Explanation

1) Concept:

Using the Rolle’s Theorem verify the result

2) Theorem:

Rolle’s Theorem – Let f  be a function that satisfies the following 3 hypotheses:

i. f is continuous on the closed interval [a, b]

ii. f is differentiable on the open interval (a, b)

iii. f (a)=f (b)

Then there is a number c in (a, b) such that f'c=0

3) Given:

fx=1-x2/3

4) Calculation:

Now, f-1=0=f(1) which satisfy hypothesis iii) of the Rolle’s theorem

Now differentiate f(x) with respect to x,

f'x=ddx1-x23=ddx1-d

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