Question

for these 2 theorems, pick one hypothesis, remove it from the theorem to create a new statement. then, provide a counterexample showing that the new statement is false.

(a) Thim (The Lagrange Remainder Theorem): Suppose f : 1 → R has n + 1 derivatives and ro e 1, Then for each r e I with r / ro there is some z strictly between ro and r with f(x) Pn (z) +Rn(x) where Rn is the remainder and ro)#1 Rn(z) (b) Thm (Taylor Polynomial Convergence Theorem): Let I be a neighborhood of zo and suppose f : 1 → R has all derivatives. Fix r E 1 and suppose there 31, M E Rt such that Yn E N and Ve strictly between ro and r we have mol Then rn(r)+ 0 and hence Ipn(r))- f(r)

Answer 1

g that +1 t/ 7/

d. ou 2. K. 从ーし

2 the nth onde ahere OK tfe

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