Answer
Saddle points at $(0,0)$
; Local maximum point at $f(1,1)=2$ and Local maximum point at $f(-1,-1)=2$
Work Step by Step
Given: $f_x(x,y)=4y-4x^3=0, f_y(x,y)=4x-4y^3=0$
Simplify the given two equations.
Critical points: $(0,0),(1,1),(-1,-1)$
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.
$D(0,0)=f_{xx}f_{yy}-f^2_{xy}=-16$ and $D=-16 \lt 0$
So, Saddle points at $(0,0)$
$D(1,1)=f_{xx}f_{yy}-f^2_{xy}=128$ and $D=128 \gt 0$ and $f_{xx} \lt 0$
So, Local maximum point at $f(1,1)=2$
For critical point $(-1,-1)$:
$D=f_{xx}f_{yy}-f^2_{xy}=128\implies D=128 \gt 0$ and $f_{xx} \lt 0$
So, Local maximum point at $f(-1,-1)=2$
Therefore, Saddle points at $(0,0)$
; Local maximum point at $f(1,1)=2$ and Local maximum point at $f(-1,-1)=2$
