In problem 2, the Euler Lagrange equation imply that we are looking for the function y(x) such as the surface is minimal, so the solution is a cylinder, ie y(x)=constant=c=2.
The solution to the Euler equation gives the minimal surface with is obtained when y(x) is a catenoid, y=c cosh(x/c) here c satisfies 2=c cosh(1/c). The Euler equation gives the minimal distance when the function F is the length of the arc.