Answer
Saddle Point of $f(0,1)=-3$
Local Minimum $f(2, 1)=-19$
Local Minimum $f(-2, 1)=-19$
Work Step by Step
$$f_x(x,y)=4x^3-16x=0$$ and $$f_y(x,y)=6y-6=0$$
Critical points are: $(-2,1 )$ and $(0,1)$ and $(2,1)$
Apply the second derivative test for $(-2,1 )$ and $(0,1)$ and $(2,1)$
$D=f_{xx}f_{yy}-f^2_{xy}=(-16)(6)-0=-96-0=-96 \lt 0$
$D(2,1)=f_{xx}f_{yy}-f^2_{xy}=(-16)(6)-0^2=192-0=192 \gt 0$
$D(-2,1) =f_{xx}f_{yy}-f^2_{xy}=(32)(6)-0^2=192-0=192 \gt 0$
So, Local Minimum $f(-2, 1)=-19$
Local Minimum $f(2, 1)=-19$
Saddle Point of $f(0,1)=-3$
