# Gradient, critical points, and second derivatives

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?

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

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