f[x_,y_]:={3-3x-3(x^2)+3y+3x y-(y^2),-2+x+(x^2)-3y-x y+(y^2)};
g[u_,v_]:={-1+3u-3(u^2)+2v+u v+3(v^2),-2-u+(u^2)+2v+u v+(v^2)}

f[3,1]

$Post=MatrixForm

D[g[u,v],{{u,v},1}]/.{u->-22,v->5}

D[f[x,y],{{x,y},1}]/.{x->3,y->1}

Det[(({{140, 10}, {-40, -10}})).(({{-18, 10}, {6, -4}}))]

Det[(D[g[u,v],{{u,v},1}]/.{u->-22,v->5}).(D[f[x,y],{{x,y},1}]/.{x->3,y->1})]

F[x_,y_,u_,v_]:={2(u^2)+x+u (x^2)-(x^3)-3y-3(u^2)y+7,-(v^3)+3u x+u v x-2(u^2)y+7}

D[F[x,y,u,v],{{x,y,u,v},1}]/.{x->0,y->1,u->-2,v->-1}

Det[(({{1, -15}, {-4, -8}}))]

-Inverse[(({{1, -15}, {-4, -8}}))].(({{4, 0}, {8, -3}}))

$Post=.

f[x_,y_,z_]:=-15+2x-(x^2)+4y-(y^2)+2z-(z^2)

Solve[D[f[x,y,z],{{x,y,z},1}]=={0,0,0}]

((x^2)/4)+((y^2)/4)+((z^2)/25)/.{{x->1,y->2,z->1}}

Lagrange multipliers

L[x_,y_,z_,λ_]:=f[x,y,z]-λ(((x^2)/4)+((y^2)/4)+((z^2)/25)-1)

solutions=N[Solve[D[L[x,y,z,λ],{{x,y,z,λ},1}]=={0,0,0,0}]]

f[x,y,z]/.solutions

ContourPlot3D[((x^2)/4)+((y^2)/4)+((z^2)/25)==1,{x,-5,5},{y,-5,5},{z,-5,5}]

f[x_,y_,z_]:=8-2x+(x^2)-2y+(y^2)-2z+(z^2)

Solve[D[f[x,y,z],{{x,y,z},1}]=={0,0,0}]

((x^2)/25)+((y^2)/9)+((z^2)/16)/.{x->1,y->1,z->1}

L[x_,y_,z_,λ_]:=f[x,y,z]-λ(((x^2)/25)+((y^2)/9)+((z^2)/16)-1)

solutions=N[Solve[D[L[x,y,z,λ],{{x,y,z,λ},1}]=={0,0,0,0}]]

f[x,y,z]/.solutions

f[1,1,1]