The condition $f(x)f'(x)<0$ just means that whenever $f$ is positive, then $f'$ is negative, and conversely. Note that $f$ can never be zero, so it is either always positive or always negative (hence $f'$ is either always negative or always positive).
(A) and (B) are certainly not true, since one can find the counterexamples $f(x)=e^{-x}$ and $f(x)=-e^{-x}$. The first counterexample also shows that (C) is false, since $|f(x)|=f(x)$ there.
If $f$ is positive, then $f'$ is negative, hence $f$ must be decreasing. If $f$ is negative, then $f'$ is positive, hence $f$ must be increasing. But notice that in this case $|f|=-f$, hence $|f|$ is always decreasing, so (D) is true.