Question

What condition must be met by a function f for it to have a Taylor series centered at​ a?

A. The function f must have a relative extrema at point a.
B. The function f must also be able to be written as a Maclaurin series.
C. The function f must have derivatives of all orders on an interval containing the point a.
D. The function f must have a finite number of derivatives on an interval containing the point a.

Answer 1

A function must have derivatives of all orders on an interval containing the point for it to have a Taylor series centered at that point.

Step-by-step explanation:

The condition that must be met by a function f for it to have a Taylor series centered at a point a is that the function must have derivatives of all orders on an interval containing the point a. This means that not only does the function itself need to be continuous, but all its derivatives must be continuous as well, at least within the interval around a. In other words, for a Taylor series to exist for a function, it must be infinitely differentiable at the point of expansion. This means that options A, B, and D are not correct, as they refer to relative extrema, Maclaurin series, or a finite number of derivatives, none of which are requirements for a Taylor series. Instead, option C correctly captures the requirement for a Taylor series.