I am trying to maximize the following function:
$$ \max f(x) = (x\cdot B_{1})/(A_{1}+x)\ -\ (x\cdot B_{2})/(A_{2}+x) $$
I know that the following is true: $x > 0 \; \land \; \frac{A_{1}}{B_{1}} \leq \frac{A_{2}}{B_{2}} $. I have tried with various numerical libraries and plotting solutions to try to maximize it, it seems to hold. I, however, am unsure if it would be possible to maximize the function without using numerical techniques, would this be possible, and if so how?