Consider the function $g:\left(0,1\right)\rightarrow\mathbb{R}$ defined by $ g\left(x\right)=\left(1-x\right)\left(1-\frac{1}{1+f\left(x\right)}\right), $ where $f\left(x\right)$ is a continuously differentiable function that is positive and strictly increasing with $x\in\left(0,1\right)$. Can one claim that if $g$ has a maximum, this maximum is unique?
Thanks in advance,
Paul