## Calculus (3rd Edition)

$f(x)-g(x)$ is non-increasing.
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 $x \geq 0$.