Let us consider we need to find a curve between $(-x,y)$ to $(x,y)$ such that the surface of revolution of this curve has minimum surface area. I proceeded to find area by considering the infitesimal length segment $dS$ and multiplying it with $y$ gives us $dA$. Integrating it from $-x$ to $x$ gives us $A$. After doing all the necessary math, i got the path as, $y= \operatorname{acosh}(x/a)$, where a is a constant. Now when $x/y$ ratio is calculated, when $x/y<0.66$, we have two extremums, one being minima and another the maxima. At $x/y= 0.66$, we only get one solution and when $x/y> 0.66$, no stationary solution is observed. Coming to the question, when, $0.51< x/y < 0.66$, the surface of revolution of this curve is the minimum and doesnt depend on $x$ coordinate.
Its the Goldschmidt curve which obviously doesnt come to us because we implicitly assume that whatever curve we're finding is sufficiently differentiable. Does this mean that every such question posed about paths has a Goldschmidt kind of path, which doesnt come to us in solution, but exists, which is the path of least action for us?