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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.

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

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.

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.

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.

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.