Convergence of local Newton method for $f(x) = x^4+x^2$

Question

I am working on the following exercise:

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(x) = x^4+x^2$ with the start value $x^{(0)} \ne 0$ and let $x^{(k)}$ be the sequence generated by the Newton method to minimise $f$.

1. Show that $x^{(k)} \ne 0$ for all $k$.
2. What can you say about the convergence behaviour of $x^{(k)}$ ?

COMMENT: My version of the local Newton method calculates the directions $d^k$ as the solution of $\nabla^2 f(x^{(k)})d = -\nabla f(x^{(k)})$.

We note that $\nabla f(x) = 4x^3+2x$ and $\nabla^2 f(x) = 12x^2+2.$

I have already done 1.). But I do not know what to say about 2.). As far as I know the convergence of the local Newton method depends on the choice of $x^0$.

Can we say anything more special here?

Answer 1

The hint: Newton's method gives iterations $$ x_{k+1}=x_k-\frac{4x_k^3+2x_k}{12x_k^2+2}=x_k\cdot\frac{4x_k^2}{6x_k^2+1}=x_k\cdot\rho(x_k). $$ If you manage to show that $|\rho|\le r<1$ then you are done (why?).