Is the point of Newton's Method for finding a local minimum of a function to find critical and saddle points of a function, or is it to find the true minima? I am using Newton's Method on the function $f(x)=x^4+4x^2y+4y^2+2x+2y$ starting at the point $(1,1)$ and am confused by the results I'm getting. After my first iteration, I am at $(x,y)=(1,-3/4)$ with a function value of $3/4$. I get a gradient of $0$ on my second iteration, though, so I don't move anywhere after that. Using WolframAlpha, I saw that this point is a saddle point of the function, so I didn't know if that is what Newton's Method is all about or if I'm doing something wrong.
There are smaller function values. For example, $(-2,-2)$ has a function value of $-8$.
It's rather unusual to hit a critical point exactly after one iteration of Newton's method: in this example it only happens if you start at a point of the form $(1,y)$. More typically, you might converge to a critical point as you take more and more iterations, or not converge at all: periodic or chaotic behaviour is possible. For example, there is a $2$-cycle: start at $(5/4, -29/32)$, the first iteration takes you to $(3/4, -13/32)$ and the second takes you back to $(5/4, -29/32)$.
In this particular problem, $(1,-3/4)$ is the only critical point; there are no local minima or maxima.