Function two wariables - hessian matrix is 0

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Homework Help Overview

The discussion revolves around the classification of stationary points for functions of two variables, specifically in the context of the Hessian matrix and its implications for determining extrema.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster seeks guidance on classifying stationary points when the Hessian matrix is zero, particularly in more complex functions. Some participants question the accuracy of the Hessian being zero and suggest that it may not be the case for all entries. Others inquire about the relationship between the Hessian and the classification of stationary points.

Discussion Status

The conversation is exploring different interpretations of the Hessian matrix's role in classifying stationary points. Some participants provide insights into the conditions under which stationary points can be classified as minima, maxima, or saddle points based on the eigenvalues of the Hessian. There is an acknowledgment of the need for further analysis, such as Taylor expansion, when encountering null eigenvalues.

Contextual Notes

Participants note that the original poster may have misunderstood the properties of the Hessian matrix and its implications for classification. There is also a reference to standard calculus resources that outline the conditions for classifying stationary points.

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Homework Statement


what can I do if I have hessian = 0? ex. function
[tex]f(x,y)=x^2+y^4[/tex]
hessian is 0, what now? this is simply but what can i do in more complicated functions?
 
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The hessian matrix of f isn't zero in all entries, if you do the math you can easily see that.

However, i don't understand your question. I'm guessing you wish to classify the stationary points of f?
 
sorry, I didnt explain it good:) i need to know how can I classify stationary to extreme points or non-ekstreme depending on hessian?
 
player1_1_1 said:
sorry, I didnt explain it good:) i need to know how can I classify stationary to extreme points or non-ekstreme depending on hessian?

The demonstration for why these conditions are the conditions that allow you to classify stationary/non stationary points is in every calculus book that approaches this subject.

After you get the hessian matrix, you have to calculate it's eigen values, if all of them are positive then you have a minimum, if all are negative, then it is a maximum, if some are negative and some are positive then you don't have any of the previous one's, in this case you might have a saddle point (imagine a horse saddle-like surface).

In the case that you at least one null eigen value then, to find out what kind of stationary point it is, you will most likely have to calculate the taylor expansion of the function and see how the function varies with that approximation.
 

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