Local Extrema, Hessian & Eigenvalues

In summary, the critical points for the function f(x,y,z)=x3-3x-y3+9y+z2 are (-1,√3,0). The Hessian matrix at this point has diagonal entries -6, -6√3, 2 and the eigenvalues are -6, -6√3, 2. Since there are two eigenvalues of opposite signs, this should be a saddle point. However, the model answer states that it is a local maximum, which contradicts this. It is suggested to contact the professor for clarification. Additionally, the calculation for the Gaussian curvature of a circle and torus is requested.
  • #1
kingwinner
1,270
0
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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  • #2
Please help!
?Should it be a saddle point or a local maximum?
 
  • #3
Everything you've done appears to be correct.
 
  • #4
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...
 
  • #5
I would say it should be 'saddle'. Model answers can make mistakes... why don't you try emailing your prof?
 
  • #6
OK, I think the model answer is wrong. This is a disaster...
 
  • #7
can anybody please tell me how to calculate gaussian curvature of a circle?
and of torus
 

1. What is a local extremum?

A local extremum is a point on a function where the function reaches either a maximum or minimum value within a small neighborhood of that point. It is different from a global extremum, which refers to the maximum or minimum value of the entire function.

2. What is the Hessian matrix?

The Hessian matrix is a square matrix that contains second-order partial derivatives of a multivariate function. It is used in mathematics and physics to analyze the behavior of functions at critical points, such as local extrema.

3. How is the Hessian matrix used to determine local extrema?

The Hessian matrix can be used to determine the type of critical point (maximum, minimum, or saddle point) by examining the sign of its eigenvalues. A positive-definite Hessian matrix with all positive eigenvalues indicates a local minimum, while a negative-definite Hessian matrix with all negative eigenvalues indicates a local maximum. A mix of positive and negative eigenvalues indicates a saddle point.

4. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts from linear algebra that are used in the analysis of matrices. Eigenvalues are scalar values that represent the scaling factor of an eigenvector when multiplied by a matrix. Eigenvectors are non-zero vectors that remain in the same direction when multiplied by a matrix.

5. How are eigenvalues and eigenvectors related to the Hessian matrix?

The eigenvalues of a Hessian matrix represent the curvature of a function at a critical point, while the corresponding eigenvectors indicate the direction of the curvature. This information is used to determine the type of critical point, as well as the behavior of the function near that point.

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