Local Extrema, Hessian & Eigenvalues

1. Mar 23, 2008

kingwinner

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: Mar 23, 2008
2. Mar 23, 2008

kingwinner

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

3. Mar 23, 2008

nicksauce

Everything you've done appears to be correct.

4. Mar 23, 2008

kingwinner

Hi,

Do you mean the correct answer should be "saddle"?

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

5. Mar 23, 2008

nicksauce

I would say it should be 'saddle'. Model answers can make mistakes... why don't you try emailing your prof?

6. Mar 23, 2008

kingwinner

OK, I think the model answer is wrong. This is a disaster...

7. Oct 23, 2009

tayyaba aftab

can any body plz tell me how to calculate gaussian curvature of a circle???
and of torus