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 190: 64


$f(x)-g(x) $ is non-increasing.

Work Step by Step

Consider $$y= f(x)-g(x) $$ Then $$y'= f'(x)-g'(x) $$ Since $f (x) \geq g(x)$ for all $x \geq 0$, then $y' \leq 0$ for all $x \geq 0$. Hence, $y$ is non-increasing. Since $y(0) = f (0) − g(0) = 0$, then $y(x) \leq 0 $ for all $x \geq 0$. Hence, $f (x) − g(x) \leq 0$ for all $x \geq0$, and $f (x) \leq g(x)$ for $ x \geq 0$.
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