Problem 49. Throughout this problem, let E: R R be a differentiable function such that E'(x) = E(x) for all xeR. (a) Prove that if E(xp) = 0 for some xy R, then E(x) = 0 for all x R. (b) Prove that if E(0) = 1 and F: R R is another differentiable function such that F'(x) = F(x) for all x Rand F(0) =1, then E(x) = F(x) for all x R. (c) Prove thatif E(0) = 1, then E(x+y) = E(x)E(y) for all x,y R. [Hint: define f(x) = E(x+y)/E(y) and prove that f'(x) = f(x) for all x and f(0) = 1.] (d) Assume that E(0) = 1. Define e = E(1). Prove that E(n) = * for all x R and that 2.5