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

Answer

(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.)
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.