Question: Let f(a:) be a function satisfying f(0)=0, f'(0)=2, f\"(0)=-6 and {f(3)(m){ S 18 forO S a: S 1. Find the Taylor polynomial of degree 2 of f at :1: = 0 and then find 2wf(m) m>0+ 372 - Answer: The Taylor polynomial of degree 2 of f at m = 0 is P2013) = El- Near .1: = 0 , the function f(m) is equal to 132(3) plus some remainder, that is it) = P2($) + R3013)- By the Lagrange formula for the remainder we know that f (3) (C) 3 (I: for some 0 E [0, cc]. R308) = 3! Since (3) c |f 35) _ a a. we deduce that lim 23"?\" a: El