Finding absolute extrema of a multivariable function.

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burt

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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.
1586675404229.png


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.
 
Last edited:
burt said:
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.
Click to expand...
If I were to do this problem, I would use wolframalfa to plot z = x^2 + y^2 - 4xy - and visualize boundaries
 
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.
 
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