Suppose $f(x) = G^{-1}\left(\frac{1}{2}+\frac{1}{2}G(H(x)\right)+G^{-1}\left(\frac{1}{2}G(H(1-x)\right) $,
where $x\in[0,1]$ and $G,H$ are strictly increasing, strictly concave functions with $G(0)=H(0)=0$ and $G(1)=H(1)=1$.
If we assume first-order differentiability of $f(.)$, then I believe it is easy to show that the maximiser is some $x>0.5$. However, I am having some difficulties in showing that this is a unique maximiser.
I've drawn a few examples in Mathematica but I cannot seem to construct an example for falsification which supports my intuition that the maximiser is unique, but unfortunately, this lacks any proper rigor.
Any ideas would be most grateful, thanks!