We can write some FOCs which describe optimal point. But how to find point which satisfies these FOCs?

I suspect it is not necessary to find this point, you can get correct answer without knowing exact location of it. I haven't solved this problem yet, so it is a hypotesis only...

Haven't ACed yet, but it seems I found the algo which uses 2 binary searches only.

If you want to use BS, you should have a sorted structure. What is it? Do you serach this point on some fixed line?

Of course, I convert initial problem to some other problem, and there it is ordered structures =)

Convex structure also gives O(lg n).
For example golden proportion search om segment.