Q55. Let f(x,y)=x^3+y^3+3xy for all real x and y. Then there exist distinct points P and Q such that f has a...

Is there a faster method other than using the "2nd derivative test"?

Usual method:

Solve to get (x,y)=0 or (x,y)=(-1,-1).

Then check the sign of D=36xy-9, and

For x=0, D<0, so it is saddle point.

For x=-1, D>0 and so it is a maximum.

Just curious if there is a faster and easier way out for this question?

Thank you very much.