Numerical Analysis: Given a function and successive derivatives at one point, what's the value of the function at another point?

Question

Example of an exercise I'm trying to solve: Find the value of $f ( 4)$ given that $f (6 )=350 , f ' (6 )=87 , f'' (6 )=30 , f ''' (6 )=4$ and all other higher derivatives of $f (x) at x=6$ are zero.

I don't know how to proceed.

Answer 1

More hypothesis are required. If f is analytic, then $$f(x)=\sum_{n=0}^\infty{f^{(n)}(6)\over n!}(x-6)^n=\sum_{n=0}^3{f^{(n)}(6)\over n!}(x-6)^n.$$ But if $f$ isn't analytic (even if is $C^\infty$) the values of the function and all the derivatives in one point give zero information about the function in other point.

Answer 2

Write the Taylor series: $f(4)=f(6)+(4-6)f'(6)+\frac 12(4-6)^2f''(6)+\dots$ No guarantees of accuracy, but if the function is analytic you will get it exactly.