Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises: 6


(a) $f$ is decreasing on $(1,5)$ and $(7,8)$ and increasing on $(0,1)$ and $(5,7)$. (b) $f$ has a local minimum at $x=5$ and 2 local maximums at $x=1$ and $x=7$.

Work Step by Step

(a) We use the Increasing/Decreasing Test: Therefore, looking at the graph, we would look for the intervals where $f'$ is negative and positive. We see that on $(1,5)$ and $(7,8)$, $f'\lt0$. Therefore, $f$ is decreasing on these two intervals. On $(0,1)$ and $(5,7)$, $f'\gt0$. So $f$ is increasing on these two intervals. (b) We use The First Derivative Test: Here, $f'$ changes from negative to positive at $x=5$, so $f$ has a local minimum at $x=5$. Also, $f'$ changes from positive to negative at $x=1$ and $x=7$, so $f$ has 2 local maximums at $x=1$ and $x=7$. (Do not let the fluctuations in this graph confuse you. If you look at the definitions, you see that only the sign of $f'$ and whether it changes sign or not matters.)
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