Use an iteration to approximate the specified cubic root

The cubic $x^3-6x+1=0$ has a root $\alpha$ in $(0,1)$.

(i) Find $\alpha$ to 5 decimal places.1 mark

(ii) For $x_{n+1}=(x_n^3+1)/6$, $x_1=0$, use a calculator to determine the sequence’s behaviour.2 marks

(iii) Prove algebraically that, if the sequence converges to $\alpha$, its limit is the root of the given cubic.2 marks