Chapter 1.2, Problem 28E

a.

To determine

To show: $x=0$ is the only fixed point of the function $g(x)=x+x^3$ .

b.

To determine

To Prove : If $0<x_0<1$ then $x_0<x_1<x_2<\mathrm{...}$

c.

To determine

To show: FPI fails to converge to a fixed point, while $g\text{'}(0)=1$ . Together with the condition if $x_0>x_1>x_2>\mathrm{...}$ FPI may converge to a fixed-point r or diverge from r when $\left|g\text{'}(r)\right|=1$ .