A function is called absolutely monotone if $f$ and all derivatives are non-negative. Your definition means that $f'$ is absolutely monotone. The main theorem about such functions is Bernstein's theorem which says that every absolutely monotone function on an interval $(a,b)$ is a Laplace transform of a (non-negative) measure. This implies that such a function is analytic in the strip $a<\Re z<b$.
So if an absolutely monotone function has an absolutely monotone extension on
a larger interval, this extension is always unique, by uniqueness theorem for analytic functions.

EDIT. If the signs of derivatives alternate, $(-1)^nf^{(n)}\geq 0$ then $f(-x)$ is absolutely monotone.

References: D. V. Widder, Laplace transform, Princeton NJ, 1941.