University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 4 - Section 4.1 - Extreme Values of Functions - Exercises - Page 215: 13


The table in Exercise 13 matches with graph $(d)$.

Work Step by Step

Here we see that $f'(a)$ does not exist but $f'(b)$ exists and equals $0$. In graphs $(b)$ and $(c)$, both the points at $x=a$ and $x=b$ are local minimum or maximum of the graph; and since the graph at these points are smooth, $f'(a)$ and $f'(b)$ must both exist and equal $0$, according to Theorem 2. In graph $(a)$, though the sharp point at $x=a$ indicates that $f'(a)$ does not exist, yet the sharp point at $x=b$ also indicates the same thing, which here $f'(b)=0$. Therefore, only graph $(d)$ correctly portrays the table in Exercise 13.
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