It would help if you could say what course you've encounterd this problem in, so to know what theorems you're reasonably expected to be aware of.

Also, where have you gotten on this probelm?

Last week I suggested a proof of how meridians are geodesics. Given two points on sphere, you can align the sphere so that it's a surface of revolution where the great circle connecting those two points is a median.

Meridians onf surface of revolution.

You'd need to add some Riemannian geometry to fully prove distance minimizing. Show that all great arcs are geodesics, and visa versa. Then show that conjugate points are antipodal points, so that geodesics are length minimizing up until another one equals it in length between two points - which again, via great arcs, doesn't happen until antipodal points. From that, you'll have shown what the cut locus of a point is.

Also, if you're taking some kind of mathematical methods class, analysis, pdes, or calculus of variations, then the Euler-Lagrange equation is perhaps how they expect you to solve the problem.