Find and Classify all Critical Points of a Multivariable Function

Find critical points by solving for all points that make the first partial 0. Classify those critical points using the Hessian matrix.
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Hi and thanks a lot for your help! My problem is the following: I would like to draw a phase diagram for a system of 3 differential equations And it has three parameters

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Can you help please how classification of critical points of system in three equation in 3d

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[@nabihahsafiyahmohdazizan4995]

How to know 3:06 ? Like how did you get y=0 or y=2?

[@NakiaRimmer]

I should have been more clear, I am setting the expression 3y(y-2) equal to zero since critical points occur when the first partial derivatives are zero simultaneously.

[@ndivhomusehane3296]

what happened to 3x^2 when you moved to 3y^2-6y

[@NakiaRimmer]

That was under the case where x = 0

[@jacoblutz6211]

Why is it that when you analyze the sign of D(x,y) and it becomes >0, that you always analyze fxx. Why not fyy or fxy or even f(x,y). What's the reason for this?

[@NakiaRimmer]

Great question! I’m going to throw out a bunch of terms and give you a link to a video. If that video doesn’t do it for you, then throw the terms into the search and hopefully you can find a better one.

[@NakiaRimmer]

For a function f(x,y), there is a matrix made up of the second partial derivatives. That matrix is called the Hessian. The determinant of this matrix along with the eigenvalues of the matrix are used to classify critical points.

[@NakiaRimmer]

The determining factor that distinguishes a local max from a local min is the calculation of whether the matrix is positive definite or negative definite. Checking the sign of f_xx is a shortcut to this calculation.

[@NakiaRimmer]

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