A function f has derivatives of all orders for all real numbers, x. Assume f(2) = −3, f ′(2) = 5, f ″(2) = 3 and the third derivative of f at x = 2 is −8. 1. Write the 3rd degree Taylor polynomial for f about x = 2 and use it to approximate f(1.5).

2. The fourth derivative of f satisfies the inequality f(4) x ≤ 3 for all x in the interval [1.5, 2]. Use the LaGrange error bound on the approximation to f(1.5) found in part (a) to explain why f(1.5) cannot equal −5.