Chapter 4, Problem 9RE

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using Rolle's Theorem In Exercises 9-12, determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f ' ( x ) = 0 . If Rolle's Theorem cannot be applied, explain why not f ( x ) = x 3 − 3 x − 6 ,     [ − 1 , 2 ]

To determine

To calculate: The value of c such that f'(c)=0 with the help of Rolle’s Theorem for f(x)=x33x6

Explanation

Given:

The function f(x)=x3âˆ’3xâˆ’6 in the interval [âˆ’1,2].

Formula used:

Rolleâ€™s Theorem states that for any function g(x) that is differentiable in the interval (a,b) and continuous in the interval [a,b], if g(a)=g(b), then there exists c in the interval (a,b) such that f'(c)=0.

Calculation:

Consider the provided function:

f(x)=x3âˆ’3xâˆ’6

Clearly, the function being a polynomial is continuous and differentiable on the whole number line.

For Rolleâ€™s Theorem to be applicable the value at the end points needs to be equal.

f(âˆ’1)=(âˆ’1)3âˆ’3(âˆ’1)âˆ’6

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