Let f(x) be a function that is continuous and differentiable at all real numbers, and f(3)=5, f'(3)=-2, f''(3)=4, f'''(3)=-1. Also, |f''''(x)| <= 6 for all x in the interval [3,3.2]

Question

1. Write a third order taylor polynomial for f(x) about x =3 to approximate f(3.2). [I got 4.6787]

2. Find an interval [a,b] where a <= f(3.2) <= b [I'm not sure how to solve this]

Answer 1

This article should help.

mathinsight. org/achieving_desired_tolerance_taylor_polynomial_desired_interval_refresher