Consider the function g(x) = 2x(1 − x). x = 0 and x = 1/2 are fixed points of g(x).

(a) Why should we expect that fixed point iteration, starting even with a value very close to zero, will fail to converge toward x = 0?

(b) Why should we expect that fixed point iteration, starting with p0 ∈ (0, 1) will converge toward x = 1/2? What order of convergence should we expect?

(c) Perform seven iterations starting from an arbitrary p0 ∈ (0, 1) and numerical confirm the order of convergence.