Assignment:
The fourth degree Taylor Polynomial for f(x) about x = 3 is :
P4(x) = -1 - (x-1)^2 + 2(x-1)^3 - 4(x-1)^4.
i) Find f^(n)(1) for n = 1, 2, 3, 4.
ii) Show that the function has a local maximum at x = 1.
iii) The fifth derivative of f(x) satisfies |f^(s)(x)| less than or equal to 25000 for all x in [0.8, 1.1].
Use the Lagrange Error Formula to show that Rs(x) is less than or equal to 1/480.
iv) Use part (iii) and P4(x) to find an upper and a lower bound on f(1).
(Hint : Use the fact that f(x) = Pn(x) + Rn(x)).
Provide complete and step by step solution for the question and show calculations and use formulas.