I have been trying to solve the following minimization problem using Newton's method:
$\min _{\boldsymbol{x} \in \mathbb{R}^{2}} f(\boldsymbol{x})=x_{1}^{4}-2 x_{2} x_{1}^{2}+x_{2}^{2}+x_{1}^{2}-2 x_{1}+5$
Using the starting point $\boldsymbol{x_0}=[1,4]^T$ the method converges in one iteration as contour plot with iterates
If I use this starting point $\boldsymbol{x_0}=[-3,2]^T$
the method converges in 6 iterations as
What is the difference of these starting points. Why does the method converge in one iteration for the first starting point, but not for the second?