Let I be a neighborhood of x0 and assume that the function f : I → R is such that f (3)(x) is continuous on I and f (3)(x) > 0 for all x ∈ I. Show that if h not equal to 0 satisfies x0 + h ∈ I, then there is a unique real number λ = λ(h) such that f(x0 + h) = f(x0) + f '(x0)h + f ''(x0 + λ) h square/2 holds. Find also the limit lim h→0 λ(h) /h . Justify your answer.