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.