MHB Min-Max over a closed bounded region

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The discussion centers on finding the absolute minimum and maximum values of the function f(x,y) = x² + y² - xy over the bounded region defined by |x| + |y| ≤ 1. The user questions the validity of the book's answers, which state that the maximum occurs at points (1,-1) and (-1,1), while the minimum is at (0.5,-1) and (-1,0.5), all of which lie outside the specified region. The user expresses confusion, suggesting that the answers may pertain to a different problem. There is a consensus that either the book contains errors or the user has referenced the wrong question. The discussion highlights the importance of verifying solution validity within defined constraints.
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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