/sci/ - Science & Math

>>9487681
In the single-variable case, how do you find a relative optima? You set the first derivative equal to zero. What this means geometrically is that, at the optima, there is almost no movement up or down if you deviate from that point a bit.
To find whether that point is a maximum point or a minimum point, you use the "Second Derivative Test". If the second derivative is positive, that means your function at the optima is concave-up, so your optima is a minimum. Likewise, if the second derivative is negative at the optima, you have a concave-down graph and thus a relative maxima.

Extending this to the multivariable case, we see that we can make similar arguments. You first find the critical points of the function (set both [math]f_x = f_y = 0 [/math]). Now, you want to find whether the critical points you found are relative max, relative min, or in the case in pic related, neither.
The [math]f_{xx}[/math] term tells you the concavity in the [math]x[/math] direction. Likewise, The [math]f_{yy}[/math] term tells you the concavity in the [math]y[/math] direction.
From my understanding, the square of [math] f_{xy}[/math] tells you the concavity in the neighborhood of your point.

So what does this mean? If the "total concavity" (determinant of the Hessian matrix: https://en.wikipedia.org/wiki/Hessian_matrix)) is negative, then the function couldn't agree on whether it was concave up or down, so it's a saddle point.
If it's positive, then you know that at least it's a min or a max. point, but to see which one it is, you ned to look at the sign of [math] f_{xy} [/math].

Hopefuly this helped a bit. For what it's worth, Khan academy has a really good article on this:
https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/optimizing-multivariable-functions/a/second-partial-derivative-test