I am attempting to solve the Brachistochrone curve problem, hence I read about some calculus of variation. My understanding to Euler-Lagrange equation is: If $I(x) = \int_a^b f(x,y,y')dx$ the stationary 'points' satisfies the equation.
I recall that a single value function can obtain its minimum at boundary points, stationary points and singular points. How should I check the singular points and boundary points?