# Hessian matrix question.

• AndreTheGiant
In summary, the conversation discusses the use of a Hessian matrix in proving that x^y > y^x for certain values of x and y. It clarifies the notation and terminology used and considers the applicability of the Hessian matrix in this case. It concludes that the Hessian matrix can be used to show that there are no real local or global critical points in the given function, and therefore, the function diverges.

## 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

## 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.

AndreTheGiant said:

## 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

## 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.

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

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?

AndreTheGiant said:
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

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.

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

## 1. What is a Hessian matrix?

A Hessian matrix is a square matrix of second-order partial derivatives, commonly used in multivariate calculus. It is used to analyze the curvature of a function and determine whether it is a minimum, maximum, or saddle point.

## 2. How is the Hessian matrix calculated?

The Hessian matrix is calculated by taking the second-order partial derivatives of a function with respect to each of its variables and arranging them in a matrix form.

## 3. What is the significance of the Hessian matrix?

The Hessian matrix plays a crucial role in optimization problems by helping to determine the critical points of a function and their nature (maximum, minimum, or saddle point). It is also used in the second-order Taylor series expansion of a function.

## 4. Can the Hessian matrix be used for functions with multiple variables?

Yes, the Hessian matrix can be used for functions with multiple variables. In this case, the Hessian matrix will be a square matrix with the same number of rows and columns as the number of variables in the function.

## 5. Are there any limitations to using the Hessian matrix?

Yes, there are some limitations to using the Hessian matrix. It can only be used for functions that have continuous second-order partial derivatives. It also does not provide information about the behavior of a function beyond the second-order derivatives.