University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Section 4.2 - The Mean Value Theorem - Exercises - Page 223: 15

Answer

$f(x)$ is not continuous over the closed interval $[0,1]$, so the theorem does not apply.

Work Step by Step

Rolle's Theorem requires - that $f(x)$ is continuous on $[a,b]$ - that $f'(x)$ is defined on $(a,b)$ Observing the right endpoint of $[0,1],\quad $ $f(1)=0$ but $\displaystyle \lim_{x\rightarrow 1^{-}}f(x)=1$, so f is not left-continuous at the right endpoint. This means that f is not continuous over the CLOSED interval $[0,1]$, so the theorem does not apply.
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