At which point does $f(x) = x^{3} + 3 x^{2} - 9 x + 2$ have a local minimum?

Explanation

1. 01

   $f'(x) = 3 \left(x - 1\right) \left(x + 3\right)$.

2. 02

   $f'$ changes from negative to positive at $x = 1$, so $f$ falls then rises there: a local minimum.