Answer
There is no critical point for $f$. Hence, there is no local maximum, no local minimum nor saddle point.
Work Step by Step
We have $f\left( {x,y} \right) = \left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$.
The partial derivatives are
${f_x} = {{\rm{e}}^{{x^2} - {y^2}}} + 2x\left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$
$ = \left( {1 + 2{x^2} - 2xy} \right){{\rm{e}}^{{x^2} - {y^2}}}$
${f_y} = - {{\rm{e}}^{{x^2} - {y^2}}} - 2y\left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$
$ = - \left( {1 + 2xy - 2{y^2}} \right){{\rm{e}}^{{x^2} - {y^2}}}$
${f_{xx}} = 2x{{\rm{e}}^{{x^2} - {y^2}}} + \left( {4x - 2y} \right){{\rm{e}}^{{x^2} - {y^2}}} + 4{x^2}\left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$
$ = 2{{\rm{e}}^{{x^2} - {y^2}}}\left( {3x + 2{x^3} - y - 2{x^2}y} \right)$
${f_{yy}} = 2y{{\rm{e}}^{{x^2} - {y^2}}} + \left( { - 2x + 4y} \right){{\rm{e}}^{{x^2} - {y^2}}} + 4{y^2}\left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$
$ = 2{{\rm{e}}^{{x^2} - {y^2}}}\left( {3y - 2{y^3} + - x + 2x{y^2}} \right)$
${f_{xy}} = - 2y{{\rm{e}}^{{x^2} - {y^2}}} - 2x{{\rm{e}}^{{x^2} - {y^2}}} - 4xy\left( {x - y} \right){{\rm{e}}^{{x^2} - {y^2}}}$
$ = - 2{{\rm{e}}^{{x^2} - {y^2}}}\left( {x + y + 2{x^2}y - 2x{y^2}} \right)$
To find the critical points we solve the equations ${f_x} = 0$ and ${f_y} = 0$:
${f_x} = \left( {1 + 2{x^2} - 2xy} \right){{\rm{e}}^{{x^2} - {y^2}}} = 0$
${f_y} = - \left( {1 + 2xy - 2{y^2}} \right){{\rm{e}}^{{x^2} - {y^2}}} = 0$
Since ${{\rm{e}}^{{x^2} - {y^2}}} \ne 0$, we obtain two simultaneous equations
(1) ${\ \ \ }$ $1 + 2{x^2} - 2xy = 0$ ${\ }$ and ${\ }$ $ - 1 - 2xy + 2{y^2} = 0$
Adding the two equations of equation (1) gives
$2{x^2} + 2{y^2} - 4xy = 0$
${x^2} + {y^2} - 2xy = 0$
${\left( {x - y} \right)^2} = 0$
The solution is $y=x$.
Subtracting the two equations of equation (1) gives
$2{x^2} - 2{y^2} + 2 = 0$
${x^2} - {y^2} + 1 = 0$
Substituting $y=x$ in the last equation gives $1=0$. Thus, there is no solution. Therefore, we conclude that there is no critical point for $f$. Hence, there is no local maximum, no local minimum nor saddle point.
