Suppose that the conditions for the Mean Value Theorem hold for the function f : [a, a + h] → R, so that for some θ ∈ (0, 1) we have f (a + h) − f (a) = hf ′ (a + θh). Fix f and a, and for each non-zero h write θ(h) for a corresponding value of θ. Prove that if f ′′ (a) exists and is non-zero then lim(h→0) θ(h) = 1/2 . I have no clue how to handle this problem. Could anyone please give me some hints?