Why Must the Hessian Matrix Be Symmetric at a Critical Point?

[AndreTheGiant]

Homework Statement

Given a function f: R^2 -> R of class C^3 with a critical point c.

Why CANNOT the hessian matrix of f at point c be given by:

1 -2
2 3

Homework Equations

The Attempt at a Solution

So first i want to clarify this.

When it says f: R^2 -> R, that means the function is of two variables (x and y)?

And when it says class C^3 that means the third derivative of the function exists and is continuous. So would a function be x^3 or x^4? the third derivative would be 24x for x^4 and is continuous. The third derivative of x^3 would be 6.

I'm not sure about the answer..

 

[Ray Vickson]

Homework Statement

Given a function f: R^2 -> R of class C^3 with a critical point c.

Why CANNOT the hessian matrix of f at point c be given by:

1 -2
2 3

Homework Equations

The Attempt at a Solution

So first i want to clarify this.

When it says f: R^2 -> R, that means the function is of two variables (x and y)?

And when it says class C^3 that means the third derivative of the function exists and is continuous. So would a function be x^3 or x^4? the third derivative would be 24x for x^4 and is continuous. The third derivative of x^3 would be 6.

I'm not sure about the answer..

IF your matrix A above was a Hessian, what would the number a(2,2) = -2 represent? What would the number a(2,1) = +2 represent?

RGV

 

[AndreTheGiant]

Ah ok. I think i got it. In the hessian which is given by

fxx fxy

fyx fyy

fxy is not equal to fyx which should be the case for mixed partials?

 

[Ray Vickson]

Ah ok. I think i got it. In the hessian which is given by

fxx fxy

fyx fyy

fxy is not equal to fyx which should be the case for mixed partials?

Yes, exactly.

RGV

 

[viliperi]

I have a question considering the applicability of Hessian matrix.

So, Can I use Hessian to prove that x^y > y^x whenever y > x >= e.

At first I start by multiplying by ln() => y*ln(x) > x*ln(y)

Is it enough, if I take g(x,y) such that g(x,y) = y*ln(x) - x*ln(y) > 0 and show det(H(g)) < 0 whenever y > x >= e?

My purpose with this is to show that there are no real local or global critical points in g(x,y) when y > x >= e, and conclude that x^y - y^x diverges. I am not sure if I can use Hessian to draw that kind of conclusion.

 

[lanedance]

Hi Viliperi, welcome to PF, please start a new thread if you have a question as opposed to resurrecting an old one. You're more likely to get an anser that way as well - thanks