Calculus: Early Transcendentals 8th Edition

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

Chapter 4 - Section 4.1 - Maximum and Minimum Values - 4.1 Exercises - Page 284: 46


$f$ has 14 critical numbers.

Work Step by Step

$f'(x) = \frac{100~cos^2~x}{10+x^2}-1$ To find the critical numbers of $f$, we need to find the values of $x$ where $f'(x) = 0$ or $f'(x)$ is undefined. We can see that $f'(x)$ is defined for all values of $x$ We can find the values of $x$ where $f'(x) = 0$: $f'(x) = \frac{100~cos^2~x}{10+x^2}-1 = 0$ $\frac{100~cos^2~x}{10+x^2}= 1$ $100~cos^2~x = 10+x^2$ The left side of the equation has a maximum value of $100$ The right side of the equation is greater than $100$ when $x \lt -10$ or $x \gt 10$ Therefore, to find the points where $f'(x) = 0$, we can graph the function $f'(x)$ in the interval $[-10, 10]$ to find the points where $f'(x)$ crosses the x-axis. On the sketch of the graph of $f'(x)$, we can see that the function $f'(x)$ has $14$ x-intercepts. Therefore, $f$ has 14 critical numbers.
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