Finding absolute extrema of a multivariable function.

[burt]

I was given the following problem:
Find the absolute extrema of the function $f(x,y)=x^2+y^2-4xy$ on the region bounded by $y=x$, $y=-3$ and $x=3$.
This is my work so far.



What I'm having trouble with is figuring out the endpoints for my region. In order to know if I actually found a maximum, I need the endpoints.
Also, I'm not sure where to go from here. Do I take the derivative with respect to y of my $j(y)$ function?
I know that at the end I can compare all my answers to find my maximum and minimum. I'm just not sure how to find the rest of my candidates.

 

[Deleted member 4993]

I was given the following problem:
Find the absolute extrema of the function $f(x,y)=x^2+y^2-4xy$ on the region bounded by $y=x$, $y=-3$ and $x=3$.
This is my work so far.
View attachment 17846


What I'm having trouble with is figuring out the endpoints for my region. In order to know if I actually found a maximum, I need the endpoints.
Also, I'm not sure where to go from here. Do I take the derivative with respect to y of my $j(y)$ function?
I know that at the end I can compare all my answers to find my maximum and minimum. I'm just not sure how to find the rest of my candidates.

If I were to do this problem, I would use wolframalfa to plot z = x^2 + y^2 - 4xy - and visualize boundaries

 

[burt]

If I were to do this problem, I would use wolframalfa to plot z = x^2 + y^2 - 4xy - and visualize boundaries

How do I visualize those boundries?

 

[Deleted member 4993]

How do I visualize those boundries?

The boundaries are all flat planes - should be easy to visualize!

 

[HallsofIvy]

Actually this problem is in two dimensions so the boundaries are lines and the region is a triangle. The endpoints are the three points where y= -3 and x= 3 intersect, where x=3 and y= x intersect, and where y= -3 and y= x intersect.