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. 

Here is a link: http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00226580X
The theorem is stated in section 5. Actually it is much stronger than I stated.

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][1]. Bull. Amer. Math. Soc. 49, (1943). 178–191.


  [1]: http://projecteuclid.org/euclid.bams/1183505036