6. Let f be a function whose domain contains some open interval I. (a) Suppose f is twice-differentiable on I. i. Let a ( I. Define the functions o, p : I - R by o(t) = f(a)-If(t) + f'(t) . (a -t)], p(t) = @-t for any te I. Let b E I. Suppose b a. Show that there exists some n strictly between a, b such that o(b) = p(b)f" (n). ii. Hence deduce that for any ce I, for any x E I \\ {c}, there exists some strictly between c, r such that f() = f(c) + f'(c) . (x - c) + f (1x) 2 (x - c)?. (b) Suppose f is thrice-differentiable on I. Let a ( I. Define the functions 6, p : I - R by p (t ) = f (a) - f (t ) + f' (t) . (a - t) + 1 (1) 2 (a - 1)2 p(t ) = (a - t)3 6 for any te I . Let b E I. Suppose b a. Show that there exists some & strictly between a, b such that o(b) = p(b)f" (8)