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I am studying for my Analysis final and came across this problem I just can't get my head around:

1. The problem statement, all variables and given/known data

Find the critical points of [itex]f(x,y) = (x^2+y^2-4)(x+y)[/itex] and their nature.

2. Relevant equations

[itex]\vec{\nabla} f(x,y) = \vec{0}[/itex]

3. The attempt at a solution

[itex]\frac{\partial f(x,y)}{\partial x} = 3x^2+2xy+y^2-4 = 0[/itex]

[itex]\frac{\partial f(x,y)}{\partial y} = 3y^2+2xy+x^2-4 = 0[/itex]

These are both ellipses with center (0,0) and angles [itex]\frac{5\pi}{8}[/itex] and [itex]\frac{\pi}{8}[/itex] respectively.

According to Wolfram Alpha, there are four intersection points. (Click here to view) However, I need to solve this problem analytically and I wish to understand how it is done.

I have tried to eliminate a variable from the equations, but it is not possible due to the [itex]2xy[/itex] term. Another solution would be to write the ellipses in a parametric form, but this I feel would be overly complicated.

Could anyone please shed some light on this?

Thanks in advance!

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# Homework Help: Find the critical points of (x^2+y^2-4)(x+y) and their nature

**Physics Forums | Science Articles, Homework Help, Discussion**