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.