At which point does $f(x) = x^{3} - 12 x + 4$ have a local minimum?

Explanation

1. 01

   $f'(x) = 3 \left(x - 2\right) \left(x + 2\right)$.

2. 02

   $f'$ changes from negative to positive at $x = 2$, so $f$ falls then rises there: a local minimum.