Find the local maximum and minimum values and saddle point(s) of the function

Question

Of this function: $f(x,y)=x^2+xy+y^2+2y$.

More specifically, I'm a little confused as to how you'd find the local max and min values along with the saddle points if there are any. How would I go about doing this?

Answer 1

First you find $f_{x} (x,y)= 2x+y $ and $f_{y} (x,y)= x+2y+2$. And now $2x+y=0$ and $x+2y+2=0$ . The solution for the system is $(2/3, -4/3) $. Now use the second derivative criterion $$ d= f_{xx} f_{yy}-(f_{xy})^2$$ $f_{xx}(x,y)=2$, $ f_{yy}(x,y)=2$ and $f_{xy}(x,y)=1$. So $d= 3>0$ and $f_{xx}(x,y)=2 >0$. Theres a minimum in $(2/3, -4/3) $.