f(x,y)=x3+y3-9x2-12y+10.
fx=∂f/∂x=3x2-18x; fy=∂f/∂y=3y2-12.
fx=0⇒3x(x-6); fy=0⇒3(y-2)(y+2) at critical points.
x=0 and 6; y=2 and -2.
4 critical points: (0,2), (0,-2), (6,2), (6,-2).
fxx=∂2f/∂x2=6x-18; fyy=∂2f/∂y2=6y.
fxy=fyx=∂2f/xy∂y=0; D(x,y)=fxx.fyy-(fxy)2=fxx.fyy. If signs of fxx and fyy differ, we have a saddle-point (mixed curvature).
D(0,2)=(-18)(12)<0; D(0,-2)=(-18)(-12)>0 (max because fxx<0); D(6,2)=(18)(12)>0 (min because fxx>0); D(6,-2)=(18)(-12)<0.
Saddle-points at (0,2,f(0,2))=(0,2,-6) and (6,-2,f(6,-2))=(6,-2,-82); maximum at (0,-2,f(0,-2))=(0,-2,26); minimum at (6,2,f(6,2))=(6,2,-114).