Answer
Saddle point at $(1,-1)$; Local minimum at $(0,0)=0$
Work Step by Step
Given: $f_x(x,y)=12x-6x^2+6y=0, f_y(x,y)=6y+6x=0$
Simplify the given two equations.
Critical points: $(0,0)$ and $(1,-1)$
In order to solve this problem we will have to apply Second derivative test that suggests the following conditions to calculate the local minimum, local maximum and saddle points of $f(x,y)$ or $f(x,y,z)$.
1. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \gt 0$ and $f_{xx}(a,b)\gt 0$ , then $f(a,b)$ is a local minimum.
2. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \gt 0$ and $f_{xx}(a,b)\lt 0$ , then $f(a,b)$ is a local maximum.
3. If $D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 \lt 0$ , then $f(a,b)$ is a not a local minimum or local maximum but a saddle point.
$D(1,-1)=f_{xx}f_{yy}-f^2_{xy}=-36 \implies D=-36 \lt 0$
Thus, Saddle point at $(1,-1)$
$D(0,0)=f_{xx}f_{yy}-f^2_{xy}=36$ and $D=36 \gt 0$ and $f_{xx}=12 \gt 0$
Hence, Saddle point at $(1,-1)$; Local minimum at $(0,0)=0$
