(10 marks) Consider the equation f(x) = 0, where f(x) = e-ex. a. Show that f(x) has a unique root r, and find its multiplicity p b. Let {} C. m≥1 be the Newton sequence with initial guess ₁ = 0. Compute 20 iterates and verify the linear convergence of the sequence. Show that the convergence of {x} is indeed linear. d. Let {} m21 be the sequence generated by the modified Newton's algorithm ₁ = 0, +12-P = f(z) 'f'(z)' n = 1, 2,... where p is the multiplicity of the root r. Verify that the convergence of quadratic. e. Prove that the convergence of {z}, is indeed quadratic. {n} 21