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When I see that U shape structure, Increases both side but optimal at some middle structure.. I almost assume its ternary search..

I never learned ternary search before but I solved today's E question with binary search (my submission: 80368801).

When I iterated over the final target height I noticed that the total cost was decreasing and then increasing, so I suppose that a function with this property can be minimized with ternary search.

As far as I know, it is used to find the only maxima/ minim of a concave/ convex graph traced out by an evaluation function on various inputs. here is a detailed solution of codeforces problem, trying to explain the ternary search in more deatail.

Binary search is for monotonic functions. Ternary search is for unimodal functions. It must be strict unimodal else the search will halt as soon as it finds a plateau (0 slope). Note every strictly monotonic function is a degenerate unimodal function (with no left or right tail) so ternary search will work when binary search does as long as the monotonicity is strict. Here is some code: https://mirror.codeforces.com/blog/entry/43440

Notice the logic for determining which side to recurse on is really a difference quotient, a discrete derivative, slope etc.