A Function F(x) Satisfies the Following Conditions:
$~~~~~~~~$

Essay

A function f(x) satisfies the following conditions:

f(1) = 0, $~~~~~~~~$ f⁽ⁿ⁾ (1) = n for n = 1, 2, and 3,
3x ≤ f⁽⁴⁾ (x) ≤ 9x provided x ≥ 1.

(a) What is the Taylor polynomial p₃ (x) of degree 3 for f(x) about x = 1?
(b) What is the approximate value for f(3/2) supplied by (x)?
(c) Based on the bounds for f⁽⁴⁾ (x) given above, what is the smallest interval that you can be sure
contains the value f(3/2)?
(d) Based on your answer to (c), what is the best approximation you can give for f(3/2)?