How do you determine the behavior of critical points when you have the Hessian?

In summary, the conversation discusses using the gradient and Hessian to find critical points and determine if they are local maxima, minima, or saddle points. The process involves calculating the determinant and trace of the Hessian matrix and checking the values of the diagonal. A saddle point is identified if the inequality fxx*fyy-fxy^2 < 0 is satisfied, while a local maximum or minimum is determined based on the signs of fxx and fyy.
  • #1
kelp
9
0
Hello,
I have solved for the critical points using the gradient, and I have solved for the Hession, which yields a 2x2 matrix. I have plugged in my critical points into the gradient.

Now, do I apply the same rules as in linear algebra where I find the determinant and trace to calculate positive definite, negative definite, and indefiinite? I currently add up the diagonal and check the determinant. If they are both positive, then it's a local min, if they are both negative, it's a local max, and a saddle point if none of the above, correct?
 
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  • #2
if fxxfyy-fxy^2 < 0, saddle point

if fxxfyy-fxy^2 > 0 then

local maximum if fxx,fyy < 0 and local minimum if fxx,fyy > 0
 

1. How do I calculate critical points using the Hessian matrix?

The first step in determining critical points using the Hessian matrix is to find the first-order partial derivatives of the function. Next, calculate the second-order partial derivatives and use them to form the Hessian matrix. Finally, set the determinant of the Hessian matrix equal to 0 and solve for the critical points.

2. What is the significance of the Hessian matrix in determining critical points?

The Hessian matrix provides important information about the curvature and behavior of a function at a given point. It contains the second-order partial derivatives, which can help determine whether a critical point is a minimum, maximum, or saddle point.

3. Can the Hessian matrix be used to classify all critical points?

No, the Hessian matrix can only be used to classify critical points in two-dimensional functions. In higher dimensions, more complex methods are needed to classify critical points.

4. How can I determine the behavior of a critical point using the Hessian matrix?

The behavior of a critical point can be determined by analyzing the eigenvalues of the Hessian matrix. If all eigenvalues are positive, the critical point is a minimum. If all eigenvalues are negative, the critical point is a maximum. If the eigenvalues are a mix of positive and negative, the critical point is a saddle point.

5. Are there any limitations to using the Hessian matrix to determine critical points?

Yes, the Hessian matrix can only be used for twice-differentiable functions. If a function is not twice-differentiable at a critical point, the Hessian matrix cannot be used to determine its behavior. Additionally, the Hessian matrix may not provide enough information to classify critical points in higher dimensions or for more complex functions.

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