For this problem, there's actually 4 distinct problems. I have no idea what to do for these. Would you have to do something called the hessian matrix,which I am confused by. Can someone run me through one of these problems step by step so I can do the other three on my own?
Thanks!!
Sure. Take $f(x)=x^2+xy+y^2$. First, we find the partial derivatives with respect to $x$ and $y$.
$\frac{\partial }{\partial x}f(x,y)= 2x+y.$
$\frac{\partial }{\partial y}f(x,y)= 2y+x.$
From this, we can find the critical points.
Then we simply use this formula:
$D(x,y)=f_{xx}f_{yy}-(f_{xy})^2.$
When you plug your critical points into $D(x,y)$,
If $D>0$ AND $f_{xx}<0$, then this is a relative maximum.
If $D<0$ AND $f_{xx}>0$, then this is a relative minimum.
If $D<0$, then this is a saddle point.
If $D=0$, then the test is inconclusive.
I hope this helps!