Any inflexion-point solutions to Euler-Lagrange equation?

[Happiness]

The following pages use Euler-Lagrange equation to solve for the shortest distance between two points and in the last paragraph mentions: "the straight line has only been proved to be an extremum path".

I believe the solution to the Euler-Lagrange equation gives the total length $I$ a stationary value and not an extremum value, so should the book have said: "the straight line has only been proved to be an extremum path or an inflexion-point path"?

Also, Professor Susskind, I believe, never mentions inflexion-point solutions when he teaches Euler-Lagrange equation, but only extremum and saddle-point solutions.

 

[Orodruin]

Yes, the EL equations give stationary solutions. The prime example of this is finding stationary pathlengths on the sphere. The global minimum between two points is the shorter part of the great circle connecting them. The long part lf the same great circle can be shown to be neither a minimum or a maximum.