Why must $f$ be non-negative and $f'$ be negative? The following is the Theorem 6.17 of the book Apostol's Analytic Number Theory:

My question is that I don't understand why is it necessary for $f$ to be a non-negative function and $f'$ to be negative for all $x\ge x_0$? (esp. it has not been supposed the function to be non-negative in Abel's identity, i.e. Theorem 4.2)  

In the use of Abel's identity the author makes use of  the estimate
$$\left|\int_x^y A(t) f'(t)dt\right|\leq \sup|A| \int_x^y  |f'(t)|dt=C\int_x^y  (-f'(t))dt= C(f(x)-f(y))$$
For the middle we need $f'$ non positive. And finally we want to take the $y\rightarrow \infty$ limit so we want that $f(y)=O(f(x))$ for all $y>x$. This is best achieved if $f$ is positive.
