Is there any theoretical upper bound on the second derivative of a twice-differentiable function?

Question

Lets assume that f(x) is a twice-differentiable and nonlinear function, where x is bounded by the interval l ≤ x ≤ u, and the function itself is bounded by L ≤ f ≤ U. We know the values of l, u, L and U. At the moment, we also know f(l), f([l+u]/2) and f(u). My question is, how do we find a parameter K so that |f′′(x)|≤ K in the whole interval of l ≤ x ≤ u?

Answer 1

No, you can achieve any value on the second derivative - in general you can just write any function and integrate it twice.

In this case that doesn't quite work but still it's very easy to construct some sharply curving bump under these conditions.