Tylor theorm

please read carefully all Question posted in pictures and show me all a detailed answers for each questions so I can be able to understand how do you solve them step by step and write it in two ways as a page attached and as the theorem

From this book

http://introduction-to-real-analysis-4th-ed.pdf

Thanks for your advance

Taylor’s Theorem. Let , let and let be such that and its
derivatives are continuous on I and such that exists on If then
for any in I there exists a point between and such that
Proof. Let and be given and let denote the closed interval with
endpoints and . We define the function on by
for
(1) Then,
If we define on by
(2) For ,
then
(3) An application of Rolle’s Theorem yields a point between
such that
(4) Hence, we obtain
(5)
(6) which implies the result.
(1) HAND IN. Supply the details for each of the
six steps of Taylor’s Theorem as listed on the link
below:
Taylor’s Theorem
Note: “Supply the details” means exactly that, i.e.,
clearly JUSTIFY each step of the proof. This might
take you 20 minutes or it might take you 20 hours.
(1) HAND IN. Supply the details for each of the
six steps of Taylor’s Theorem as listed on the link
below:
Taylor’s Theorem
Note: “Supply the details” means exactly that, i.e.,
clearly JUSTIFY each step of the proof. This might
take you 20 minutes or it might take you 20 hours.