Hi I'm practicing finding extremas/monotonicity.
What if the derivative changes sign (function changes the monotonicity) at some point $x$ that is outside the domain?
Like, for example, $f(x) = x^3e^{\frac{-1}{x}}$ the domain of derivative is $D_{f'} = (\infty;0),(0,\infty)$ and one of derivative's solutions is $x = 0$ which does not belong to neither domain of function $D_{f}$ nor domain of derivative $D_{f'}$. The function changes monotonicity, but does it have extreme value in there?
And similiar situation: What if the $x$ does belog to domain of $f$ but does not belong to domain of $f'$?
It doesn't: if a function doesn't exist at a certain point, it doesn't have a derivative there either.
How can $x=0$ be a zero of the derivative, if the derivative doesn't even exist in $x=0$...?
That's possible since a function isn't necessarily differentiable at every point of its domain. That doesn't mean it can't have an extreme value at such a point, think of $|x|$ at $x=0$ for example: no derivative, but the function has a minimum there.