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!

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- Mar 12th 2010, 01:43 PM #1

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- Mar 12th 2010, 05:02 PM #2
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.

- Mar 12th 2010, 05:11 PM #3

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