## [SOLVED] Limit points

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,

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

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

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

This is the sequence...
$x^0 = 1.01$
$x^1 = -1.003650$
$x^2 = 1.001647$
$x^3 = -1.000787$
$x^4 = 1.000385$
$x^5 = -1.000190$
$x^6 = 1.000094$
$x^7 = -1.000047$
$x^8 = 1.000023$
$x^9 = -1.000011$
$x^{10} = 1.000005$

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

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.