Let f(x,y)=x3+y3−3axy.
At the critical values of the function, both the first derivatives are zero.
⇒ (x,y) = (a,a) represents a minimum if a>0 and represents a maximum if a<0.
When (x,y)=(a,a),f(x,y)=x3+y3−3axy=−a3.
If a=0, the function becomes x3+y3, which does not have any maximum or minimum and has a saddle point at (x,y)=(0,0).
Therefore, we conclude as under:
1. If a>0, the function does not have a maximum but has a local minimum at (a,a) having a value −a3.
2. If a<0, the function does not have a minimum but has a local maximum at (a,a) having a value −a3.
3. If a=0, the function does not have either a maximum or a minimum.