A constructive converse of the mean value theorem

Consider the following converse of the Mean Value Theorem. Let f be a differentiable function on [a, b]. If c e (a, b), then there are a and ß in [a, b] such that (f(ß) - f(a))/(ß - a) = f'(c). Assuming some weak conditions to be mentioned in Section 3, Tong and Braza [3] were able to prove this statement. Unfortunately their proof does not provide a method to compute a and ß. We give a constructive proof.