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I was thinking about the sign of the n th Derivative.

More precisely real analytic for x>0 where the expansion point Goes from 0 to + oo.

Sine and cosine are simple.

But for other cases , when the expansion point Goes to + oo it is less trivial.

For instance - too give a quite general case -
What about e.g.

D^a(n) f(b(m)) < 0
D^a2(n) f(b2(m)) > 0.

What are the solutions and how to find them easily ?
Maybe this is easy sorry.
I havent thought about it much.