Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 4: Applications of Derivatives - Section 4.4 - Concavity and Curve Sketching - Exercises 4.4 - Page 213: 72

Answer

See graph and explanations.

Work Step by Step

Step 1. The graph of $f''(x)$ indicates that the function will be concave up on $(-\infty,c)$ ($f''\gt0$, $c$ is the zero of $f''(x)$) and concave down on $(c,\infty)$ ($f''\lt0$). The zero of $f''(x)$ ($x=c$) indicates an inflection point of the function. Step 2. The graph of $f'(x)$ indicates that the function decreases in regions (left and right sides) where $f'(x)\lt0$ and increases in the middle region where $f'(x)\gt0$. The zeros of $f'(x)$ indicate extrema of the function (maximum or minimum depending on if $f'(x)$ changes from positive to negative or vise versa). Step 3. Using point $P$, we can sketch the graph of the original function $f(x)$ as shown in the figure.
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