Please prove the following:

Question 7 5 marks (Proofs) Suppose f : R - R is twice-differentiable and a

Question 7 (Proofs) Suppose f : R 9 R is twice-differentiable and a < x are two fixed real numbers. (a) By applying Rolle's theorem to the function h(t) := f (t) f (a) a) (x a)2 show that 2 for some Cl e (a, x). (b) By an application of the mean value theorem, conclude that f"(c)(x 2 for some c e (a, x). 5 marks Remark: This is a proof of the n = 1 case of the Lagrange remainder theorem, and a slight modification yields an inductive proof for arbitrary n e N.