Finding a stationary point using Newtons Method for two variables.

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So I've written a function in MATLAB which solves a non-linear system of equation:

$f(x,y) = 0$

$g(x,y) = 0$

for two variables.

I am now tasked with finding a stationary point (critical point) for a function f(x,y) using my newton method program.

How is this supposed to work? How is the newton method helpful with finding criticals points? I can't really see how to apply this.

I should also add, that for this specific problem I am given the function:

$f(x,y)= (x-1)^2 + (y-1)^2$

Since my function for the Newton method relies on two functions f & g as input arguments.

Should I then treat any function f as a two functions?

i.e:

$f(x,y)= (x-1)^2$ and $g(x,y)= (y-1)^2$?