EDITED. The following theorem of Bernstein answers the question:
If $f$ is infinitely differentiable on an interval and no derivative changes sign, then $f$ is analytic.
Your condition that all derivatives are monotone of course implies that none of them changes sign. Therefore, if such a function is extended on a larger interval with preservation of the property that no derivative changes sign, then such an extension is unique.
S. Bernstein, Sur la définition et les propriétés des fonctions analytiques d'une variable réelle, Math. Ann. vol. 75 (1914) pp. 449-468.
A survey of the later results on the topic is
Polya, G. On the zeros of the derivatives of a function and its analytic character. Bull. Amer. Math. Soc. 49, (1943). 178–191.
