Local Extrema, Hessian & Eigenvalues

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Homework Help Overview

The discussion revolves around finding and classifying critical points of a multivariable function, specifically focusing on the Hessian matrix and its eigenvalues. The original poster expresses confusion regarding the classification of a critical point as a saddle point versus a local maximum based on the eigenvalues derived from the Hessian matrix.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the classification of critical points using the Hessian matrix and its eigenvalues, questioning the apparent contradiction between their findings and the model answer. Other participants express similar confusion and seek clarification on the classification.

Discussion Status

Participants are exploring the classification of the critical point, with some suggesting that the model answer may be incorrect. There is a general consensus among some participants that the point should be classified as a saddle point based on the eigenvalues, while others are uncertain and seek further guidance.

Contextual Notes

Participants are discussing the implications of having eigenvalues of opposite signs and the potential for errors in model answers. There is also a mention of a separate question regarding the calculation of Gaussian curvature, indicating a broader interest in related topics.

kingwinner
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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.
=================

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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Please help!
?Should it be a saddle point or a local maximum?
 
Everything you've done appears to be correct.
 
nicksauce said:
Everything you've done appears to be correct.
Hi,

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

But the answer says that it's a local max...I am confused...
 
I would say it should be 'saddle'. Model answers can make mistakes... why don't you try emailing your prof?
 
OK, I think the model answer is wrong. This is a disaster...
 
can anybody please tell me how to calculate gaussian curvature of a circle?
and of torus
 

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