The example, from my homework, is to find the critical points of $f(x,y)=(x^2 + y^2)e^{-(x^2+y^2)}$. I can bash it out, but I want to be confident I'm seeing the most creative way to solve it. This just begs to be restated somehow in polar form. WolframAlpha even tells me that the critical points lie at the origin and on the unit circle.
If I make the substitution, the problem becomes incredibly simple. In polar form - $f(r, \theta) = r^2e^{-r^2}$ - it doesn't even depend on $\theta$, so the problem is just simple single-variable calculus. Is this a valid argument? Why or why not? How does it hold up in general?
Note that in polar form,
$$f(r) = r^2e^{-r^2}$$
and set,
$$\frac{df}{dr} = e^{-r^2}\cdot 2r(1-r^2) = 0$$
which yields $r=0$, hence the origin.