Note to poster: The given inequality should read "|f^(4)(x)| ≤ 3".
Otherwise, the derivative may not be bounded below.
The Lagrange error bound for the n-th degree Taylor polynomial centered at 'a' is given by
|En(x)| ≤ M|x - a|^(n+1)/(n+1)!, where M is an upper bound for |f^(n)(x)| for all points between a and x.
In our case, a = 2, x = 1.5, n = 3, and M = 3 (via |f^(4)(x)| ≤ 3).
So, |E₃(1.5)| ≤ 3|1.5 - 2|³⁺¹/(3+1)! = 1/128.
In other words, the error in using the third degree Taylor polynomial of f(x) centered at x = 2 to estimate f(1.5) is no bigger than 1/128.
I hope this helps!