Given an nth order Taylor polynomial of f(x) about a number b, the error estimate is given as

provided that

.

If given a greatest bound for the error I can figure out an interval J containing b such that the error is less than or equal to the bound.

Is there any way to find an "optimal" value for M, in the sense that we find an interval J such that the error takes its maximum possible (given) value somewhere in the interval. (I guess this would also mean, "How can we find the largest possible interval J?")

I think I'm being clear here. If not I'll give an example to work with to make things clearer.

-Dan