This is a small part of a much larger question to do with optimization but is the only part I cant get!

Apply Newton's method (without linesearch) to minimize the univariate function,

$\displaystyle f(x) = \frac{11}{546}x^6 - \frac{38}{364}x^4 + \frac{1}{2}x^2$, $\displaystyle x \in \mathbb{R}$,

starting from $\displaystyle x^0 = 1.01$ and let $\displaystyle {x^k}$ be the generated sequence of iterates.

Prove that the limit points of the sequence of iterates $\displaystyle {x^k}$ are +1 and -1 as $\displaystyle k \rightarrow \infty$.

This is the sequence...

$\displaystyle x^0 = 1.01 $

$\displaystyle x^1 = -1.003650 $

$\displaystyle x^2 = 1.001647 $

$\displaystyle x^3 = -1.000787 $

$\displaystyle x^4 = 1.000385 $

$\displaystyle x^5 = -1.000190 $

$\displaystyle x^6 = 1.000094 $

$\displaystyle x^7 = -1.000047 $

$\displaystyle x^8 = 1.000023 $

$\displaystyle x^9 = -1.000011 $

$\displaystyle x^{10} = 1.000005 $

And Newtons method is... $\displaystyle x^{k+1} = x^k - \frac{f'(x^k)}{f''(x^k)}$

So how would I do this. Some examples in class just had the sequence of iterates with the statement "clearly the sequence converges to ..." but I think I should be giving a proper proof.