Function two wariables - hessian matrix is 0

[player1_1_1]

Homework Statement

what can I do if I have hessian = 0? ex. function
$$f(x,y)=x^2+y^4$$
hessian is 0, what now? this is simply but what can i do in more complicated functions?

 

[Gunthi]

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?

 

[player1_1_1]

sorry, I didnt explain it good:) i need to know how can I classify stationary to extreme points or non-ekstreme depending on hessian?

 

[Gunthi]

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.