On finding the critical points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

Question

Given some constant $a\in\mathbb{R}-\{0\}$, find $x_0$ such that $f'(x_0)=0$ where $$f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x.$$

I have managed to write $f'(x_0)=0$ as $$\small\dfrac{a(ax_0^2-1)}{x_0(ax_0^2-a^2x_0+1)}\left(\dfrac{ax_0^2-a^2x_0+1}{ax_0}\right)^a\\=\small\left(\ln\left(\dfrac{a^2x_0^2-x_0+a}{ax_0}\right)+\dfrac{a(ax_0^2-1)}{a^2x_0^2-x_0+a}\right)\left(\dfrac{a^2x_0^2-x_0+a}{ax_0}\right)^x$$

Note that for the simplest case when $a=1$, we have that $x_0=1$ and it is a point of inflexion.

I suppose there isn't a closed form for $x_0$ based on elementary functions, but I expect Lambert's $W$ function to be of use in the manipulation of $x_0$ as the subject of the equation above.

P.S. I asked this question on MSE a few months ago and have offered multiple bounties but I was not able to gather very useful observations.

Answer 1

Here is a plot (obtained using numerical methods in Maple) of the solutions for $0 < a < 1$.

EDIT: Here's the code, as requested.

f:= (x-a+1/(a*x))^a - (1/x-1/a+a*x)^x;
plots[implicitplot](diff(f,x),a=0..1,x=0..3,gridrefine=3);