Optimization convergence requirements

Hello I have a function from [0,1] to [0,1], which I can shown to always have a unique maximum. I have been using Optim.jl with the default Brent’s method to compute it. I am a statistician, not a specialist on numerical algorithms. But I still need to confirm that this algorithm will never fail me. I see a reference to Brent’s book and I guess I will have to read parts of it. However, I would like some pointers to where I can find criteria for garanteed convergence. Further, I can also comput...