Min-Max over a closed bounded region

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SUMMARY

The discussion centers on finding the absolute minimum and maximum values of the function \(f(x,y)=x^{2}+y^{2}-xy\) over the closed bounded region defined by \(\left | x \right |+\left | y \right |\leq 1\). The user expresses confusion regarding the book's provided answers, which state that the maximum occurs at points (1,-1) and (-1,1), while the minimum occurs at (0.5,-1) and (-1,0.5). The user asserts that these points do not lie within the specified region, suggesting a potential error in the book or a misunderstanding of the problem.

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Yankel
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Hello again

I have another question regarding absolute min-max over a region. This is a weird one.

My function is:

\[f(x,y)=x^{2}+y^{2}-xy\]

and the region is:

\[\left | x \right |+\left | y \right |\leq 1\]

Now, I have plotted the region using Maple:

View attachment 2601

The answer in the book where it came from is weird, it say that the maximum value is at the points: (1,-1) and (-1,1) while the minimum is at (0.5,-1) and (-1,0.5)

All these points are NOT in the region ! Am I missing something ?

My intuition say that these answers are for a different question.

Thanks !
 

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There is certainly something wrong here. The answer bears no relation at all to the question. Either the book is wrong or you have looked up the answer to the wrong question. :p
 
No, these answered were copied from a book, not by me, so the wrong answer was copied.

Just wanted to make sure, I thought I miss something very basic :D
 

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