Computing bounds on a function over an interval

Background:
I want to find all zero crossings of a function, expanded over a basis. I decided I'd give IntervalRootFinding.jl a shot. In essence, IntervalRootFinding.roots needs to evaluate the function (and its derivative) on an interval x and expects an interval y back which holds the minimum and maximum values attained by the function (derivative) on x.

Now, since my function is expressed as a linear combination of basis functions, my best idea so far is to find the corresponding interval y_i for the ith basis function, multiply this interval by the expansion coefficient (which will flip the interval if the coefficient is negative) and sum all the minima and maxima. We cannot simply use the union of the intervals since two functions on top each other would necessarily increase the interval. At the same time, for two disjoint functions, adding the intervals would make the bound looser than it necessarily needs to be. Is there a way to find a tighter bound?

How would the interface for finding the extrema of each basis function over an interval look like? If B isa Basis, with continuous first dimension, extrema(B, interval) could e.g. return a vector of intervals, but I am not sure if this is the best way. It would also be dependent on which particular implementation of Interval one chooses (IntervalSets.jl, IntervalArithmetic.jl, etc.).

Finally, is there a more clever way to find roots than this?

Start with the issue's discussion of IntervalRootFinding.roots, Basis, and extrema(B, interval), then compare how IntervalSets.jl and IntervalArithmetic.jl represent intervals. Clarify whether tighter bounds and a root-finding approach are needed, and define an agreed interface and completion criteria before implementing anything.