Numerically approximating critical points of a multivariable function

[burt]

I was given the following question: Numerically approximate all critical points of the function \(f(x,y)=2y(x+2)-x^2+y^4-9y^2\).
I cannot figure out how to do this. I tried using a CAS calculator, but ended up with an error.
I know that the critical points are when either partial derivative is undefined or when both partial derivatives are equal to zero. I just cannot figure out how to get those numbers with this function.
I tried graphing it on a handheld ti-nspire cx ii cas and it didn't come up. I tried graphing it on geogebra and got something very weird looking (attached). Can anyone guide me on how to proceed?

 

[HallsofIvy]

The partial derivatives are so easy here there would be no reason to try a "numerical" derivative method. \(\displaystyle f_x= 2y- 2x= 0\) and \(\displaystyle f_y= 2x+ 4+ 4y^3= 0\). From 2y- 2x= 0, y= x so 2x+ 4+ 4y^3= 4x^3+ 2x+ 4= 0. Use a numerical method to solve that cubic equation. Using "Desmos Graphing Calculator", https://www.desmos.com/calculator, I get approximately x= -0.835

 

[burt]

The partial derivatives are so easy here there would be no reason to try a "numerical" derivative method. \(\displaystyle f_x= 2y- 2x= 0\) and \(\displaystyle f_y= 2x+ 4+ 4y^3= 0\). From 2y- 2x= 0, y= x so 2x+ 4+ 4y^3= 4x^3+ 2x+ 4= 0. Use a numerical method to solve that cubic equation. Using "Desmos Graphing Calculator", https://www.desmos.com/calculator, I get approximately x= -0.835

I think you missed a -18y.
Does that change the way to solve it?

 

[Steven G]

Of course it does as the two equations are now different. So what progressed have you made? Can we see your work so we know what hints to give you?

 

[burt]

Of course it does as the two equations are now different. So what progressed have you made? Can we see your work so we know what hints to give you?

I think I solved it!
I just found the partial derivatives, set them equal to zero, and used the second derivative test to find the answer.

 

[burt]

Thanks for all your help!