(A) Suppose f (x) has a local maximum at c, so that f (c+ h) S f(c) for small values of h. Prove that limJLHOJr W91 s 0 and limhnor MW 2 0. Use these inequalities to conclude that f'(c) = 0. We can show similarly that if f(:r:) has a local minimum at c, then f'(c) = 0. These two results are known as Fermat's Theorem. (B) Suppose that f (x) is continuous on [a,b], differentiable on (a,b), and f (b) = f (a). Use the previous result in combination with the Extreme Value Theorem to show that f'(c) = 0 for some 0 in (a, b). This result is known as Rolle's Theorem