Answer
Saddle point at $(0,0)$
Work Step by Step
Since, we have $f_x(x,y)=-y e^ x=0, f_y(x,y)=e^y-e^x=0$
After solving the above two equations, we get:
The critical point is: $(0, 0)$
As per the second derivative test, for $(0,0)$ we have
$D=f_{xx}f_{yy}-f^2_{xy}=-1$ and $D=-1 \lt 0$
Thus, we have: Saddle point at $(0,0)$