Where does the curve $x^y = y^x$ intersect itself? This problem is quite easily solved by using logarithms and derivative and forming the function $f(x,y) = x^y - y^x$, however there are assertions that this problem can be solved without using either of the two. I can not see how one would proceed to solve this problem in the absence of logarithmic manipulation and differentiation. Can anyone shed some light on this?

Ok the function $f\colon \mathbb R^2 \to \mathbb R$ is continuous. And $f(1,2)=-1$ and $f(2,1)=1$ so it has to take the values between -1 and 1. Thus for some point we have that $f$ is zero.
