Let f be a function having derivatives of all orders for allreal numbers. The third-degree Taylor polynomial for f about x=2 isgiven by

T(x) = 7-9(x-2)2-3(x-2)3

a. find f(2) and f '' (2).

b. Is there enough information given to determine wheather fhas a critical point at x = 2?

If not, explain why not.

If so, determine whether f(2) is a relative maximum, arelative minimum, or neither, and justify your answer.

c. Use T(x) to find an approximation for f(0). Is there enoughinformation given to determine whether f has a critical point atx=0?

If not, explain why not.

If so, determine whether f (0) is a relative maximum, arelative minimum, or neither, and justify your answer.

d. The fourth derivative of f satisfies the inequality for all x in the closed interval [0,2]. Use thelagrange error bound on the approximation to f (0) found in part(c) to explain why f (0) is negative.