What is the local minimum value of $f(x) = x^{3} + 3 x^{2} - 45 x + 4$?

Explanation

1. 01

   $f'(x) = 3 \left(x - 3\right) \left(x + 5\right) = 0$ at $x = -5$ and $x = 3$.

2. 02

   $f''(x) = 6 x + 6$: $f''(3) = 24$, so $x = 3$ is a local minimum.

3. 03

   The value there is $f(3) = -77$.