Finding a critical point for a function $f(x,y)$ using Newton Raphson method

I was tasked with finding the critical point of a function f(x,y)=(x-1)^2 + (y-1)^2

Using my Newton Raphson program written in MATLAB.

    function [x0,y0] = NewtonMethod(f, g, a, b)
    
    y0 = b; 
    x0 = a;
    
    tol = 1e-18;
    
    resf = tol + 1; 
    resg = tol + 1;
    
     while  resf > tol && resg > tol
    
      dfdx = NPderiv(f, x0, y0, 1);  %function which calculates the partial derivative 
      dfdy = NPderiv(f, x0, y0, 2);
      dgdx = NPderiv(g, x0, y0, 1);
      dgdy = NPderiv(g, x0, y0, 2);
        
      jf1 = [f(x0,y0) dfdy; ...
           g(x0,y0) dgdy];
       
      jg1 = [dfdx f(x0,y0); ...
            dgdx g(x0,y0)]; 
       
      j2 = [dfdx dfdy; ...
            dgdx dgdy];
     
      jx = det(jf1)./det(j2); 
      jy = det(jg1)./det(j2); 
    
      x0 = x0 - jx; 
      y0 = y0 - jy;
     
      resf = abs(f(x0,y0));
      resg = abs(g(x0,y0));
    end
    end
    
    y0 = b; 
    x0 = a;
    
    tol = 1e-18;
    
    resf = tol + 1; 
    resg = tol + 1;
    
     while  resf > tol && resg > tol
    
      dfdx = MyNPderiv(f, x0, y0, 1);  
      dfdy = MyNPderiv(f, x0, y0, 2);
      dgdx = MyNPderiv(g, x0, y0, 1);
      dgdy = MyNPderiv(g, x0, y0, 2);
        
      jf1 = [f(x0,y0) dfdy; ...
           g(x0,y0) dgdy];
       
      jg1 = [dfdx f(x0,y0); ...
            dgdx g(x0,y0)]; 
       
      j2 = [dfdx dfdy; ...
            dgdx dgdy];
     
      jx = det(jf1)./det(j2); 
      jy = det(jg1)./det(j2); 
    
      x0 = x0 - jx; 
      y0 = y0 - jy;
     
      resf = abs(f(x0,y0));
      resg = abs(g(x0,y0));
    end
    end

My quandary is however how I should treat my second input variable "g", since they want me to use this specific code to find the critical point of $f(x,y)=(x-1)^2 + (y-1)^2$

My idea was to replace g with the linearization of f since

L(x,y) is approximately f(x,y)

Any ideas?