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Find and classify all critical points.

I am confused about the following:

The Hessian matrix is diagonal with diagonal entries 6x, -6y, 2.

Now, the diagonal entries of a diagonal matrix are the eigenvalues of the matrix. (this has to be true, it is already diagonal, so it is already diagonalized and the eigenvalues must appear on the main diagonal!!!)

(-1,√3,0) is a critical point.

The Hessian (which is diagonal) at this point has diagonal entreis -6, -6√3, 2, so the eigenvalues of the Hessian at this point are -6, -6√3, 2.

There are two eigenvalues of opposite signs, so this should be asaddle!!!

However, the model answer says that it is alocal maximum!!

But 2 is apositiveeigenvalue, so it can't be a local maximum.

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I can't understand this. Why are they contradicting? Can someone see where the mistake is?

Please let me know! Thank you!

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# Local Extrema, Hessian & Eigenvalues

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