Chapter 1.2, Problem 7E

Use Theorem 1.6 to determine whether Fixed-Point Iteration of $g\left(x\right)$ is locally convergent to the given fixed point $r$ .

(a) $g\left(x\right)={\left(2x-1\right)}^{1/3}$ , $r=1$

(b) $g\left(x\right)=\left(x^3+1\right)/2$ , $r=1$

(c) $g\left(x\right)=\sin x+x$ , $r=0$