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: 15


The function has an absolute minimum at $x=0$ but does not have any absolute maximum.

Work Step by Step

$$f(x)=|x|\hspace{2cm}-1\lt x\lt2$$ The graph is sketched below. - The graph obviously has an absolute minimum value, which is $f(0)=0$. - The graph, however, does not have an absolute maximum value. The graph does seem to reach the highest point at $x=2$, but since we consider here the interval $(-1,2)$, we do not include the point where $x=2$. And since there are no other maximum values, the graph does not have any absolute maximum in the defined interval. This answer is still consistent with Theorem 1, because Theorem 1 requires that the function $f$ in question be examined in a CLOSED interval, not an open one like in this case.
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