Solving equation of extrema with two variables

[quazzimotto]

I'm trying to solve for extrema in two variables and just getting more confused than anything else, I scanned and posted on link below to show exactly where I was getting stuck. So what are the relative max & mins of

h(x,y) = x^2-y^2-3*x*y?

http://home.comcast.net/~0server0/extrema.jpg

Thanks

 

[soroban]

Hello, quazzimotto!

Find the extrema: $\displaystyle \:h(x,y) \:=\:x^2\,-\,y^2\,-\,3xy$

Set the two partial derivative equal to zero and solve:

.$\displaystyle \begin{array}{ccccc} h_x & \,=\, & 2x\,-\,3y & \,=\, & 0 \\ h_y & = & -2y - 3x & = & 0\end{array}\;\;\Rightarrow\;\;x\,=\,0,\:y\,=\,0$

Second Partials Test:
. . $\displaystyle \begin{array}{ccc}h_{xx} & \,=\, & 2 \\ h_{yy} & = & -2 \\ h_{xy} & = & -3\end{array}$

\(\displaystyle D \:=\h_{xx})(h_{yy}) - (h_{xy})^2\:=\2)(-2)\,-\,(-3)^2\:=\:-13\;\) negative


Therefore, there is a saddle point at $\displaystyle (0,0,0)$