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: 48


$f'(-1)$ is bigger.

Work Step by Step

In this exercise, to find out which graph is $f$ and which graph is $f'$, we would look at the end part of both graphs - Graph blue: the graph keeps going down and passes the $Ox$ line. - Graph red: the graph goes up before starting to go down at the point where graph blue 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 if $f$ changes from going up to going down, it is obvious that $f'$ would pass the $Ox$ line to change from positive to negative. Therefore,we conclude that graph blue is the graph of $f'$ and graph red is the graph of $f$. That means $f'(-1)\gt0$ from the blue graph. Now look at the blue graph again. Find the point $x=1$. Again,we see that the tangent line drawn at $x=1$ would be 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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