Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$ I've just come across this function when playing with the Desmos graphing calculator and it seems that it has turning points for many values of $a$. 
So I pose the following problem:

Given $a \in \mathbb{R}-\{0\}$, find $x$ such that $\dfrac{dy}{dx}=0$ where $y=\left(x-a+\dfrac1{ax}\right)^a-\left(\dfrac1x-\dfrac1a+ax\right)^x$

As in most maxima/minima problems, we first (implicitly) differentiate it and set to $0$ to give $$\boxed{\small\dfrac{a(ax^2-1)}{x(ax^2-a^2x+1)}\left(\dfrac{ax^2-a^2x+1}{ax}\right)^a=\left(\ln\left(\dfrac{a^2x^2-x+a}{ax}\right)+\dfrac{a(ax^2-1)}{a^2x^2-x+a}\right)\left(\dfrac{a^2x^2-x+a}{ax}\right)^x} \tag{1}$$ I have no idea how to continue from here. I thought about taking logarithms, but it appears to me that the double $\ln$ in the term $\dfrac{a^2x^2-x+a}{ax}$ would only make the equation worse. 
(For the simplest case when $a=1$, the problem is easy: $x=1$ and it is a point of inflexion).

Let's try setting each of the terms to $0$:
Case $1$: $\left(\frac{a^2x^2-x+a}{ax}\right)^x=0$
$ \hspace{1cm}$ This is only possible when the fraction is zero; that is, solving $a^2x^2-x+a=0$ to get $$x=\frac{1\pm\sqrt{1-4a^3}}{2a^2}$$
Case $2$: $\left(\frac{ax^2-a^2x+1}{ax}\right)^a=0$
$ \hspace{1cm}$ This gives $$\begin{align}ax^2-a^2x+1=0&\implies a^2x^2+a=a^3x\\&\implies\left(\dfrac{a^2x^2-x+a}{ax}\right)^x=\left(\dfrac{a^3x-x}{ax}\right)^x=\left(\dfrac{a^3-1}{a}\right)^x=0\end{align}$$ 
$ \hspace{1cm}$ so for equality between LHS and RHS, we must have $a=1$. However, the equation 
$ \hspace{1cm}$ $ax^2-a^2x+1=0$ has no real solutions for such $a$; hence we reach a contradiction.
Case $3$: $\frac{a(ax^2-1)}{x(ax^2-a^2x+1)}=0$
$ \hspace{1cm}$ We have $x=\pm \dfrac1a$. Now LHS is $0$, and $$\left(\dfrac{a^2x^2-x+a}{ax}\right)^x=\left(1-\dfrac1a+a\right)^{\frac1a} \neq 0$$ 
$ \hspace{1cm}$ for $a \in \mathbb{R} - \{\phi\}$, where $\phi$ is the golden ratio. 
$ \hspace{1cm}$ Suppose that $a = \phi$. Then $x$ is forced to be $-\dfrac1a=-\dfrac2{1+\sqrt5}$, since $ax^2-a^2x+1=0$ 
$ \hspace{1cm}$ (undefined) when $x=\dfrac 1a$. This is impossible, since $y$ is only defined when $x>0$ for this 
$ \hspace{1cm}$ value of $a$!
Case $4$: $\ln\left(\frac{a^2x^2-x+a}{ax}\right)+\frac{a(ax^2-1)}{a^2x^2-x+a}=0$
$ \hspace{1cm}$ This is impossible from cases $1$ and $3$. 

UPDATE: I have provided a partial answer to my question, now with $x$ removed from it.

Any hints on how to solve $(3)=(4)$ are welcome.

Here, on MathOverflow: https://mathoverflow.net/questions/302105/on-finding-the-critical-points-of-fx-leftx-a-frac1ax-righta-left-fra

I've made some progress on the question, but I'm nowhere close to solving it.
For ease of reading, I will repeat the boxed equation below: $$\small\dfrac{a(ax^2-1)}{x(ax^2-a^2x+1)}\left(\dfrac{ax^2-a^2x+1}{ax}\right)^a=\left(\ln\left(\dfrac{a^2x^2-x+a}{ax}\right)+\dfrac{a(ax^2-1)}{a^2x^2-x+a}\right)\left(\dfrac{a^2x^2-x+a}{ax}\right)^x$$
At the end of my post above, there was a suggestion of setting $$\left(\dfrac{ax^2-a^2x+1}{ax}\right)^a=k$$ in the hope that we can find $k$ in terms of $a$, since only this has simple solutions for $x$.
The consequences of this are as follows: $$ax^2+1=a(a+\sqrt[a]k)x\tag{1}$$$$\small x=\frac{a+\sqrt[a]k}2\pm\frac{\sqrt{a^2(a+\sqrt[a]k)^2-4a}}{2a}\implies x^2=\frac{a(a+\sqrt[a]k)^2\pm(a+\sqrt[a]k)\sqrt{a^2(a+\sqrt[a]k)^2-4a}-2}{2a}\tag{2}$$
This means that using $(1)$, $$\frac{a(ax^2-1)}{x(ax^2-a^2x+1)}=\frac{a(ax^2-1)}{x(ax^2+1)-a^2x^2}=\frac{ax^2-1}{\sqrt[a]kx^2}=\boxed{\frac a{\sqrt[a]k}-\frac1{\sqrt[a]kx^2}}$$
We now deal with the natural logarithm term on the RHS, using $(1)$. Notice that $$\ln\left(\frac{a^2x^2-x+a}{ax}\right)=\ln\left(\frac{a(ax^2+1)}{ax}-\frac1a\right)=\boxed{\ln\left(a(a+\sqrt[a]k)-\frac1a\right)}$$ Hooray, we've got rid of the $x$! Similarly, we have $$\left(\frac{a^2x^2-x+a}{ax}\right)^x=\boxed{\left(a(a+\sqrt[a]k)-\frac1a\right)^x}$$
The last term next to the logarithmic term is a pain in the neck. However, $(1)$ can still be used. $$\frac{a(ax^2-1)}{a^2x^2-x+a} = \frac{a(ax^2+1)-2a}{a(ax^2+1)-x}=\boxed{\frac{a^2(a+\sqrt[a]k)x-2a}{a^2(a+\sqrt[a]k)-x}}$$
Let's combine all of the boxed terms to see what we currently have: $$k\left(\frac a{\sqrt[a]k}-\frac1{\sqrt[a]kx^2}\right)=\left(\ln\left(a(a+\sqrt[a]k)-\frac1a\right)+\frac{a^2(a+\sqrt[a]k)x-2a}{a^2(a+\sqrt[a]k)-x}\right)\left(a(a+\sqrt[a]k)-\frac1a\right)^x$$
Now we use $(2)$. The steps to the final result are horrible, so I will just give it here. Note that when I took logarithms in the RHS, I used the fact that $\ln(ab^c)=\ln a + c\ln b$. Just hope I haven't made any mistakes: $$\text{LHS}=\frac k{\sqrt[a]k}\left(1-\frac{4a-2}{a^2(a+\sqrt[a]k)^2\pm(a+\sqrt[a]k)\sqrt{a^2(a+\sqrt[a]k)^2-4a}-2}\right)\tag{3}$$ and $$\begin{align}\ln(\text{RHS})&=\frac{a(a+\sqrt[a]k)\pm\sqrt{a^2(a+\sqrt[a]k)^2-4a}}{2a}\ln\left(a(a+\sqrt[a]k)-\frac1a\right)\tag{4}\\\small&+\ln\left(\ln\left(a(a+\sqrt[a]k)-\frac1a\right)+\frac{a^3(a+\sqrt[a]k)^2\pm a^2(a+\sqrt[a]k)\sqrt{a^2(a+\sqrt[a]k)^2-4a}-4}{2a^3(a+\sqrt[a]k)-(a+\sqrt[a]k)\pm\sqrt{a^2(a+\sqrt[a]k)^2-4a}}\right)\end{align}$$ Is it even possible now to rearrange this to make $k$ the subject?

Here is a visualisation of the equation $\text{LHS}=\text{RHS}$.

