To distinguish if a critical point is a saddle point or not,. .

[alexio]

To distinguish a critical point is a saddle point or not, It is useful way to use discreminent.

The discreminent is fxxfyy-fxy^2.

What I want to know is how to prove the discreminent.

 

[Reverie]

To distinguish a critical point is a saddle point or not, It is useful way to use discreminent.

The discreminent is fxxfyy-fxy^2.

What I want to know is how to prove the discreminent.

This is the multivariate version of the second derivative test from calculus. If the second derivative is positive you are at a minimum. If the second der is Negative you are at a maximum.

Let D be the discriminant matrix, and h a 2x1 column vector. At a minimum h^T*D*h>0 for all small h indicates a minimum. This is an approximation to the function. Thus, D, which is diagonalizable since it is symmetric, is positive definite. Thus, all eigenvalues are real and positive. So, the determinant of D, which is what you call the discreminent, is positive. At a maximum, D is negative definite, which means both eigenvalues are negative. Hence, the discreminent is also positive. If D is neither positive definite or negative definite, the D has one positive and one negative eigenvalue meaning the discreminent is negative.

That's the basic idea of the proof. Hope it was helpful.

Take care,
Reverie

 

[tacman]

The discreminent is a useful tool for determining the nature of a critical point. It is defined as fxxfyy-fxy^2, where fxx is the second partial derivative of the function with respect to x, fyy is the second partial derivative with respect to y, and fxy is the mixed partial derivative.

To prove the discreminent, we can use the second derivative test. This test involves evaluating the second partial derivatives at the critical point and using their values to determine the nature of the critical point.

If the discreminent is positive, then the critical point is a local minimum or maximum. If it is negative, then the critical point is a saddle point. And if it is equal to zero, further analysis is needed to determine the nature of the critical point.

In summary, the discreminent is a valuable tool for distinguishing between saddle points and other types of critical points. By using the second derivative test and evaluating the discreminent, we can confidently determine the nature of a critical point.