Proposition 1 Let [a, b] C I and let a be a real number strictly between Df(a) and Df(b). Then there exists some c E (a, b) such that Df (c) - a. (a) Prove that if f is a convex function, then for all x, y ER, f ( y) 2 f ( x) + Df(x) (y - x). (b) Hence or otherwise, prove that if f is convex, then there exists some c E [a, b] such that Df (c) _ f( b) - f (a) b - a Hint: Use part (a) and Proposition 1. (c) Using part (b), show that if f is convex, and Df (x) = 0 for all x E I, then f must be constant for all x e