Fastest Method for 8767 q55
Posted: Sun Jan 08, 2012 10:18 am
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 $$f_x=3x^2+3y=f_y=3y^2+3x$$ to get (x,y)=0 or (x,y)=(-1,-1).
Then check the sign of D=36xy-9, and $$f_{xx}=6x$$
For x=0, D<0, so it is saddle point.
For x=-1, D>0 and $$f_{xx}=6x<0$$ so it is a maximum.
Just curious if there is a faster and easier way out for this question?
Thank you very much.
Is there a faster method other than using the "2nd derivative test"?
Usual method:
Solve $$f_x=3x^2+3y=f_y=3y^2+3x$$ to get (x,y)=0 or (x,y)=(-1,-1).
Then check the sign of D=36xy-9, and $$f_{xx}=6x$$
For x=0, D<0, so it is saddle point.
For x=-1, D>0 and $$f_{xx}=6x<0$$ so it is a maximum.
Just curious if there is a faster and easier way out for this question?
Thank you very much.