[karthickgururaj]

Good article - but couple of points.

I'm not formally trained Mathematician, but I think this is not an analytic proof - it lacks rigour. It is more a graphical approach that appeals to the reader's intuition. I could be wrong here though.

Second, I really really thought the last graph will include y = x line and was very surprised to not see it :)

[chx]

No, the graph parts are explanation. The analysis of the derivative function maybe would be better if it were rewritten with the usual limes notations, nonetheless it's correct in the crucial conclusion of the ln x / x function is monotonically increasing (0, e) and monotonically decreasing (e, inf) and as such one of every pair of equal but distinct values must be from (0,e) the other from (e,inf). There are not many integers between 0 and e: 1 and 2. At 1, ln x / x is zero and in (e,inf) this function has a strict lower boundary of 0, it is never 0. Thus the only possible solution is 2 and it is.

[arketyp]

In general, couldn't a function be monotonically increasing in some interval and still have many points such that y = x? A curve that jiggles around the line y = x for instance.

[deredede]

We are not looking for points such that y = x but for distinct points x, x' that have the same y (f(x) = f(x'))

[chx]

So while the original problem was x^y=y^x he have rearranged this into ln x / x = ln y / y and so now are investigating the ln x / x function. Yes, the lettering could be clearer because y at some points means one variable and later on it means the value of the function.

[NotYourLawyer]

It's not totally rigorous, but it wouldn't be hard to fill in the details. This is a sketch of a proof, not the proof itself.

[karthickgururaj]

Yes, that's basically what I said and meant in the parent post. I was suggesting the article call it a "graphical" solution approach, not analytical.

And if I wasn't clear - it wasn't my intent to treat this any less "useful". If anything, it is more interesting (for me) to read about solutions that give an intuitive feel, than dry analysis.

[n4r9]

Proofs can be graphical or diagrammatic, though. Take for example this proof of the Pythagorean Theorem: https://cdn.britannica.com/43/70143-004-CCB17706/theorem-dem...

[hgsgm]

"analytical" means "calculus" here.