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: 3


a) Increasing/Decreasing Test b) Concavity Test c) See where concavity changes/Set second derivative to 0

Work Step by Step

a) To see where a function is increasing or decreasing, we use the Increasing/Decreasing Test, which states that if $f'(x)\gt0$ then the function $f(x)$ is increasing and that if $f'(x)\lt0$, the function $f(x)$ is decreasing. b) To see where a function is concave up or down, we use the Concavity Test. This states that if $f''(x)\gt0$, then the function $f(x)$ is concave up and if $f''(x)\lt0$, then the function $f(x)$ is concave down. c) Inflection points are when the concavity of a graph changes. So, to locate these points, simply find the x and corresponding y values of where the concavity changes from concave up to down or vice versa. An additional way to find inflection points is to solve for x when the second derivative is set to 0.
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