critical points for f(x,y)=x^3+y^3-9x^2-12y+10

Question

Find critical points for f(x,y)=x^3+y^3-9x^2-12y+10, and identify them as maximum, minimum, saddle point, or none.

Answer 1

f(x,y)=x³+y³-9x²-12y+10.

f_x=∂f/∂x=3x²-18x; f_y=∂f/∂y=3y²-12.

f_x=0⇒3x(x-6); f_y=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).

f_xx=∂²f/∂x²=6x-18; f_yy=∂²f/∂y²=6y.

f_xy=f_yx=∂²f/xy∂y=0; D(x,y)=f_xx.f_yy-(f_xy)²=f_xx.f_yy. If signs of f_xx and f_yy differ, we have a saddle-point (mixed curvature).

D(0,2)=(-18)(12)<0; D(0,-2)=(-18)(-12)>0 (max because f_xx<0); D(6,2)=(18)(12)>0 (min because f_xx>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).