I have a question on this problem. As you can see the solutions are given above. My question is when $N = 2$, when finding critical points. I get, that the gradient equals zero when
$$x = 0,\: y^2 +x^2 = 1$$
If $x = 0$ I can have $y = -1$ and $y = 1$ (also $y = 0$ but this is when derivative DNE) .
However I don't understand how the solution gets to $(\cos a, \sin a)$. I understand that $$\sin^2(x) +\cos^2(x)= 1$$ However, I do not know how to interpret or get from $y^2 +x^2 = 1$ to using polar coordinates.
Moreover, I'm totally lost when it calculates the maximum and minimum because I don't understand why it doesn't evaluate points like $(0,1)$, $(0,-1)\dots$
I think I need to understand how can I relate this problem with points of the form $(\cos a, \sin a)$ and from there how to calculate the critical points. Because right know I'm very confused.
Thanks !