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Find a curve with constraints

Hello,

I am trying to find the steepest curve f(x) constrained so that if we consider a segment [AB] with its edges on the curve (A = f(x=a), B = f(x=b)) and with [AB] length = L , the distance between f(x) and the AB segment is < h, with "h" and "L" known.

My maths knowledge faded away - I'm looking for some help on how to approach this - and then to see if it's something where Sage can assist on. Thanks!