About Taylor series

Question

Suppose $f(0) = 0, f'(0) = 2, f''(0) = −1$ and $|f''' (x)| ≤ 0.024$ for $0 ≤ x ≤ 2$. Estimate $f(1)$ to $4$ significant figures by using a Taylor polynomial. Compute a good bound for the absolute error. What's the point of mentioning that f'''(x)<0.024??

Answer 1

HINT:

$$f(x)=f(0)+f'(0)x+\frac12f''(0)x^2+\frac16f'''(\xi)x^3$$

is valid for $x\in [0,2]$ for some $0<\xi<2$

Answer 2

The point of that information is that you should look at the Lagrange form of the remainder.