Identifying no-roots case

Hi,
I am solving optimization problem (ODE parameters estimation) and as a result I have 4 values `param_1` , `param_2` , `param_3`, `param_4` marking minimum of `loss_func`. Now I need to find intersections of `loss_func(param_i)` with some critical level `alpha = const`. Typically `loss_func(param_i)` has parabolic shape with 2 intersections or "semi-constant shape with little bifurcations" and no intersections with `alpha`. For parabolic case Roots.jl derivative-free algorithms (order 0, 2, 8) work perfect but for "semi-constant" case it either fails to approximate derivatives or reaches `maxevals`. I am looking for roots search algorithm that can identify that "semi-constant" case and efficiently explore all the search area (1e-9,1e9) to say that there are no roots. Do you have an idea how it can be achieved?