1. ## max and min

hi
may i know that given f(x,y), then the nature of the critical points for this function would depend on the eigenvalue of what matrix and how do i find the eigenvalue for this function?

thanks!

2. Originally Posted by alexandrabel90
hi
may i know that given f(x,y), then the nature of the critical points for this function would depend on the eigenvalue of what matrix and how do i find the eigenvalue for this function?

thanks!
You don't even need the eigenvalues.

Find the grad and solve it equal to the zero matrix for $\displaystyle x$ and $\displaystyle y$.

$\displaystyle \nabla f = \left[\begin{matrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y}\end{matrix}\right] = \mathbf{0}$.

Then find the Hessian:

$\displaystyle \nabla ^2 f = \left[\begin{matrix}\frac{\partial ^2 f}{\partial x^2} & \frac{\partial ^2 f}{\partial x \partial y} \\ \frac{\partial ^2 f}{\partial y \partial x} & \frac{\partial ^2 f}{\partial y^2}\end{matrix}\right]$

and evaluate it at the critical points you found with the grad.

If the Hessian is positive definite, you have a minimum, if the Hessian is negative definite, you have a maximum.

(It's the equivalent of the second derivative test for a function of one variable).

You could check the eigenvalues though to check that it is a saddle point. It's only a saddle point if the Hessian has BOTH negative and positive eigenvalues.

3. Originally Posted by alexandrabel90
hi
may i know that given f(x,y), then the nature of the critical points for this function would depend on the eigenvalue of what matrix and how do i find the eigenvalue for this function?

thanks!

you want to study the determinant of the hessian for the function. the determinant of any matrix is equal to the product of its eigenvalues.