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 - Page 301: 29


We can see a sketch of one possible graph below.

Work Step by Step

$f'(5) = 0$ The slope of the graph is zero at $x = 5$ There could be a local maximum or a local minimum at this point. $f'(x) \lt 0$ if $x \lt 5$ The graph is decreasing on this interval. $f'(x) \gt 0$ if $x \gt 5$ The graph is increasing on this interval. $f''(2) = 0$ and $f''(8) = 0$ $x=2$ and $x=8$ are points of inflection. $f''(x) \lt 0$ if $x \lt 2$ or $x \gt 8$ The graph is concave down on these intervals. $f''(x) \gt 0$ if $2 \lt x \lt 8$ The graph is concave up on this interval. $\lim\limits_{x \to \infty}f(x) = 3$ $\lim\limits_{x \to -\infty}f(x) = 3$ There is a horizontal asymptote at $y=3$
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