Suppose f(x) is a function whose fourth derivative exists on the whole real line, (−∞,∞), and that f(2)=3, f′(2)=−1, f′′(2)=2, and f′′′(2)=−1.
((a)) Write down the third Taylor polynomial for f(x) expanded about x0=2.
((b)) Use the Taylor polynomial to approximate f(4).
((c)) Find a bound on the absolute error of the approximation using the fact that −3≤f(4)(ξ)≤5 for all ξ∈[2,4].