University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Section 2.5 - Continuity - Exercises - Page 96: 65


The statement is true.

Work Step by Step

Take a continuous function $f(x)$, which is never $0$ on the interval $[a,b]$. The Intermediate Value Theorem states a function $f(x)$ is continuous on $[a,b]$ and if $y_0$ is between $f(a)$ and $f(b)$, there exists a value of $x=c\in[a,b]$ such that $f(c)=y_0$. So because the continuous function $f(x)$ is never $0$ on $[a,b]$, $0$ is not between $f(a)$ and $f(b)$. Both of them are either greater than $0$ or less than $0$ and all of the values between $f(a)$ and $f(b)$ are the same. Therefore, $f(x)$ never changes sign on $[a,b]$. The given statement is true.
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