Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 32

Answer

$f(x) $ is decreasing on $(-\infty,5) $ and increasing on $ ( 5,\infty) $ $f(x)$ has a local minimum at $x=5$.

Work Step by Step

Given $$y=x^{2}+(10-x)^{2} $$ Since $$f'(x) =2x-2(10-x) $$ Then $f(x)$ has critical points when \begin{align*} f'(x)&=0\\ 2x-2(10-x)&=0\\ 4x-20&=0 \end{align*} Then $x= 5$ is a critical point. To find the interval where $f$ is increasing and decreasing, choose $x=0$ and $x= 6$ \begin{align*} f'(0)&=-20<0\\ f'(6)&=4>0 \end{align*} Hence, $f(x) $ is decreasing on $(-\infty,5) $ and increasing on $ ( 5,\infty) $. Hence, by using the first derivative test, $f(x)$ has a local minimum at $x=5$.
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