# Local Extrema, Hessian & Eigenvalues

1) f(x,y,z)=x3-3x-y3+9y+z2
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 a saddle!!!

However, the model answer says that it is a local maximum!!

But 2 is a positive eigenvalue, 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!

Last edited:

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???Should it be a saddle point or a local maximum???

nicksauce
Homework Helper
Everything you've done appears to be correct.

Everything you've done appears to be correct.
Hi,

But the answer says that it's a local max...I am confused...

nicksauce