How do I find the Critical points of a multi-variable function using MATlab?

[VinnyCee]

How do I find the Critical points of a two-variable function using MATlab?

I have a problem, I cannot seem to find the critical points of a two-variable function for the life of me!

The funtion $$f(x,y) = 10x^2y - 5x^2 - 4y^2 - x^4 -2y^4$$ is supposed to have six potential critical points. I have the following:

$$f_x = 20yx - 10x - 4x^3$$
$$f_y = 10x^2 - 8y - 8y^3$$

For what it's worth:

$$\nabla f_x = (20y - 10 - 12x^2) i + (20x) j$$
$$\nabla f_y = (20x) i + (-8-24y^2) j$$

$$\nabla f_x = \lambda\nabla f_y$$

$$\lambda = \frac{20y - 10 - 12x^2}{20x} = \frac{20x}{-8-24y^2}$$

I know that the potential critical points are at $$f_x = f_y = 0$$, but how do I find these using MATlab, or even on paper. How would I solve for both equations?

I just can't crack this problem!

P.S. - I have MATlab version 6.5

 

[VinnyCee]

Here are some preliminary (probably wrong) answers:

Ok, I took the first equation $$f_x = 20yx - 10x - 4x^3 = 0$$ and factored out a $$2x$$ to get $$2x (10y - 5 - 2x^2) = 0$$.

Then I solved for $$-2x^2$$ to get $$-2x^2 = 5 - 10y$$ and I substituted that into the second equation of $$f_y = 10x^2 - 8y - 8y^3 = 0$$ to get $$f_y = -5(5 - 10y) - 8y - 8y^3$$. This resolves down to $$-8y^3 + 42y = 25$$ which one can solve and get $$y = 1.898, 0.647, -2.545$$, but what do I do now?

 

[VinnyCee]

Using the supposed answers, I figured this:

Plug this $$y = 1.898, 0.647, -2.545$$ into $$f_x = 20yx - 10x - 4x^3$$ to get $$x = \pm 2.644, \pm 0.857, 0$$

Are these correct? When the $$(x, y)$$'s are plugged into $$f_x$$ and $$f_y$$ they are pretty close to zero (rounding). But for some reason I don't think this is correct. How would I check with MATlab?

 

[zanazzi78]

This may not be any help but ...

have you tried creating symbolic variables for x and y?

try:
>syms x y
>g=((10*x^2)*y)-(5*x^2)-(4*y^2)-(x^4)-(2*y^4)
and then solve for g
(I would have tyied this before posting but my MATlab has a bug and willl not recognise the syms command!)