Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - Review - True-False Quiz - Page 285: 17



Work Step by Step

If $f(x)$ is periodic , then there exist a number $k$ such that $f(x+k)=f(x)$. Differentiate both side with respect to $x$, we have $f'(x+k). (x+k)'=f'(x)$ But $(x+k)'=\frac{d}{dx}(x+k)=1$, therefore, $f'(x+k)=f'(x)$ Thus, $f'(x)$ is periodic (and its period is same as $f(x)$). Hence, the given statement is TRUE.
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