Setup: I have a function $$ f(x) =-\delta\left(1-\frac{\sigma^2}{(x/\delta)+\sigma^2}\right) +\delta\log\left(1+\frac{x}{\delta\sigma^2}\right) +2I\left(X;\sqrt{\frac{1}{(x/\delta)+\sigma^2}}\cdot X+G\right), $$ where:
- $\log$ in the natural logarithm with base $e$;
- $\delta$ and $\sigma$ are fixed constants;
- $X\sim\text{Bernoulli}(\nu)$ where $\nu$ is some fixed constant and $G\sim N(0,1)$;
- $X$ and $G$ are independent of each other;
- $I(\cdot;\cdot)$ is the mutual information from information theory.
Question: I would like to find the stationary point(s) of $f(x)$ for $x\in[0,\text{Var}(X)]=[0,\nu(1-\nu)]$. I would like to do this analytically but got stuck (see my attempt below). I also managed to produce a numerical plot.
Attempt (analytical): Our mutual information in $f(x)$ is known as the mutual information of a Gaussian channel and the derivative is provided in equation (2) of this paper. Taking derivations of $f(x)$ and setting it to zero, I got $$ \frac{\delta x}{(x+\sigma^2\delta)^2} -\mathbb{E}\left[\left(X+\mathbb{E}\left[X\,\bigg|\,\sqrt{\frac{1}{(x/\delta)+\sigma^2}}\cdot X+G\right]\right)^2\right]\left(\frac{1}{\delta}\right)\left(\frac{x}{\delta}+\sigma^2\right)^{-2} =0, $$ where the trouble is that I am unable to pull $x$ out and make it the subject of the equation (which will give me the stationary points). If this is not possible, is there a way to show that there is at least 2 stationary points?
Attempt (numerical): Plotting $f(x)$, I see a maximum point somewhere close to $x=0$.
