I have to numerically maximize, for $x\in (0,0.5)$, the following function for several different values of parameters $a>0$ and $b>0$: \begin{equation} f_{a,b}(x) = a\big(\log x + \log(1-x)\big) + b \log\Big(\dfrac{2\text{tanh}^{-1}(1-2x)}{1-2x}\Big) \end{equation} I would like to know if my numerical optimization recovers a global maximum, so I would like to know for which values of $a$ and $b$ the function is quasiconcave.
I have plotted the function for different values of $a$ and $b$ and quasiconcavity seems to hold, but I have not been able to prove it. Any help is very appreciated!