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Consider the function $ f(x,y) = sin(x^2+y^2)$.
The gradient is the vector $ [2xcos(x^2+y^2) , 2ycos(x^2+y^2) ] $.
I need to find the critical points, show that I can't use the second derivatives to determine their type, and use another tool to determine.

To show the second part is just to show that the second derivatives are 0 when I plug in the critical points.

But before that, I can't even find the points themselves, how does one solve the equations from when comparing the gradient to 0?

$ 2xcos(x^2+y^2) = 0 $
$2ycos(x^2+y^2) = 0 $

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It is $$x=0$$ or $$x^2+y^2=-\frac{\pi}{2}+2k\pi$$ or $$y=0$$ or $$x^2+y^2=-\frac{\pi}{2}+2k\pi$$

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