Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.8 Intermediate Value Theorem - Preliminary Questions - Page 86: 5

Answer

(a) sometimes true. (b) true. (c) never true. (d) sometimes true.

Work Step by Step

(a) sometimes true. Since $ f(c)=3$ and $3\notin (20,100)$, then according to IVT there is no $ c\in [1,5]$ such that $ f(c)=3$. However, $f(c)=3$ may still exist (the IVT simply does not guarantee it). (b) always true. Since $ f(c)=75$ and $75\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=75$. (c) never true. Since $ f(c)=50$ and $50\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=50$. (d) sometimes true. Since $ f(c)=30$ and $30\in (20,100)$, then according to IVT there is $ c\in [1,5]$ such that $ f(c)=30$ and $ c $ may be unique.
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