[elashri]

It is not minimum or maximum when you take derivative L = 0. You get sudden points which could be minimum or maximum or not. In cases of classical Lagrangian with V is the gravitional Field. It would be minimum. If you want to derive optics laws (snell/reflection) you will find that the path it not a minimum path. Actually you will find that the light will take the path that minimize the time it takes from point a to b.

For more complicated theorie. You always tty to start from one point (usually symmetric or equilibrium) and try to build your theory's Lagrangian. Usually this involve some inputs from experiments (i.e. Standard Model Lagrangian). If we are Lucky enough then the true theory wouldn't be too far.