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

1. Oct 18, 2007

### 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.

2. Oct 21, 2007

### Reverie

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