2. Let a < b. Let f and g be functions defined on [a, b]. f and g are continuous on [a, b] and differentiable on (a, b). If f (a) = f (b) = 0, we want to show that f (c) f (c)g(c) = 0 for some c (a, b). To prove it, we break this question into several parts. (a) Consider a new function h(x) = f (x)eg(x).What are h(a) and h(b)? (b) Can we say h is continuous on [a, b] and differentiable on (a, b)? If so, prove it. (c) Let x (a, b). Compute h(x). (d) Apply the Rolle's Theorem and prove f (c) f (c)g(c) = 0 for some c (a, b)