# How do you find the local extrema for f(x) = (x-3)^3 on (-∞, ∞)?

$f$ has no local extrema.
$f ' \left(x\right) = 3 {\left(x - 3\right)}^{2}$ is never undefined and is $0$ only at $0$, so the only critical number is $0$.
$f ' \left(x\right)$ is always positive for $x \ne 0$, so $f$ is increasing on $\left(- \infty , \infty\right)$.