Calculus: Early Transcendentals 8th Edition

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

Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises - Page 163: 47


$f''(1)$ is bigger.

Work Step by Step

To find out which graph is $f$ and which graph is $f'$, we would count the number of times each graph changes from going up to down and down to up and the number of times each graph passes the $Ox$ line. - Graph blue: 4 times changes up - down, down - up and 1 time passes the $Ox$ line. - Graph red: 3 times changes up - down, down - up and 4 times passes the $Ox$ line. We know that when $f$ is going up, $f'$ is positive and when $f$ is going down, $f'$ is negative. So the number of times $f$ changes from up to down and down to up would equal the number of times $f'$ passes the $Ox$ line (to change from positive to negative and vice versa). Therefore, looking at the number we count in graph blue and red above, we find that graph blue is the graph of $f$ and graph red is the graph of $f'$. So $f'(-1)\lt0$ from the red graph. Now look at the red graph again. Find the point $x=1$. We see that the tangent line drawn at $x=1$ would horizontal and parallel to the $Ox$ line. So the slope of that tangent line would be $0$. In other words, $f''(1)=0$ ($f''$ is the derivative of $f'$) In conclusion, $f''(1)$ is bigger.
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