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

A note about Case $1$ with $\,a:= 2^{-\frac{2}{3}}\,$ and $\,\displaystyle x\to 2^{\frac{1}{3}}\,$ .

Left side $\,=0\,$ for $\,\displaystyle x=2^{\frac{1}{3}}\,$ because of $\, ax^2-1=0\,$ and $\,\displaystyle ax^2-a^2x+1=\frac{3}{2}\ne 0\,$ .

Right side $\,=0\,$ for $\,\displaystyle x\to 2^{\frac{1}{3}}>1\,$:

$\,\displaystyle \lim_{z\to +0 \\x>0} z^x\ln z=0\,$ and therefore $\,\displaystyle \left(\frac{a^2x^2-x+a}{ax}\right)^x \ln \frac{a^2x^2-x+a}{ax} \to 0\,$ for $\,\displaystyle x\to 2^{\frac{1}{3}}>0\,$

$\,\displaystyle \left(\frac{a^2x^2-x+a}{ax}\right)^x \frac{a(ax^2-1)}{a^2x^2-x+a} = \left(a^2x^2-x+a\right)^{x-1} (ax^2-1) x^{-x} a^{1-x} = 0\,$

for $\,\displaystyle x=2^{\frac{1}{3}}\,$ because of $\,x-1>0\,$ and $\,\displaystyle a^2x^2-x+a=0\,$ and $\, ax^2-1=0\,$ .

If you like to work with recursions for e.g. $\,a>0,\,a\neq 1\,$

it can make sense to choose $\,\displaystyle z:=x+\frac{1}{ax}\,$ so that you get

$\displaystyle x=f_{1,2}(z):=\frac{z}{2}\pm\sqrt{(\frac{z}{2})^2-\frac{1}{a}}\,$ and $\,\displaystyle y=(z-a)^a-\left(az-\frac{1}{a}\right)^x\,$ .

We get $\,\displaystyle \frac{dy}{dz}=(z-a)^{a-1} - \left(az-\frac{1}{a}\right)^x \left(\frac{x}{z-\frac{1}{a^2}}+\frac{1}{2-\frac{z}{x}}\ln\left(az-\frac{1}{a}\right)\right)$

and a possible recursion with $\displaystyle z_0>\max\left(a;\frac{2}{\sqrt{a}};\frac{1}{a^2}\right)\,$ could be

$\displaystyle z_{n+1}=a+\left(\left(az_n-\frac{1}{a}\right)^{f(z_n)} \left(\frac{f(z_n)}{z_n-\frac{1}{a^2}}+\frac{1}{2-\frac{z_n}{f(z_n)}}\ln\left(az_n-\frac{1}{a}\right)\right)\right)^{\frac{1}{a-1}}\,$

with $\,f(z)\in\{f_1(z);f_2(z)\}\,$ .

For time reasons I haven't checked this recursion, sorry. But it's a try to simplify the calculations.

Note:

Because of $\,\displaystyle \frac{dy}{dx}=\frac{dy}{dz}\frac{dz}{dx}\,$ with $\,\displaystyle \frac{dy}{dx}:=0\,$ we have $\,\displaystyle \frac{dy}{dz}=0\,$ or $\,\displaystyle \frac{dz}{dx}=0\,$ .

$\displaystyle \frac{dz}{dx}=0\,$ means $\,ax^2-1=0\,$ which leads directly to my comment about Case $1$ .

This strengthens the claim that the substitution $\,\displaystyle z:=x+\frac{1}{ax}\,$ makes sense.

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 $\text{LHS}=\text{RHS}$.

This is not an answer, just a comment.

Your $f(a,x)=\left(x-a+\dfrac1{ax}\right)^a-\left(\dfrac1x-\dfrac1a+ax\right)^x$ gives us for different $a$ functions of $x$ that probably have different domains.

So, when you say "Given $a \in \mathbb{R}$" that is not at all as precise as it could be. It would be more precise if we could for every $a\in \mathbb R$ calculate the domain of $f(a,x)$, because, when you differentiated this function in order to find critical point(s), are you sure that they are in the domain of $f$, even if you succeed in calculating them?

It is natural to think that as $a$ vary continuously that then the domains of $f$ will vary continuously but it is a question of what happens for $a=0$, that is, does $\dfrac {1}{f(a,x)}$ has limit when $a \to 0$, or is discontinuous (in the sense of domains)?

I am of the opinion that finding solutions to your (1) should not be done before first calculating for which $a \in \mathbb R$ your $f$ is well-defined function of $x$, and what is the domain of $f$ for every $a \in \mathbb R$.