Necessary and sufficient conditions for $f(x) \geq g(x)$ over interval from higher-order-derivatives.

Question

Consider two real-valued functions $f, g$ over some interval $[a,b] \subset \mathbb{R}$. I would like to know whether one can give simple necessary and sufficient conditions for $f(x) \geq g(x), \forall x \in [a,b]$ in terms of higher order-derivatives of $f$ and $g$. For instance, if you give me a list of all derivatives (including the function values) of the two function evaluated at some point on the interval, can I infer whether one graph lies above the other? There should be something that can be said via Taylor expanding the two functions but I can't find anything anywhere.