What is the local maximum value of $f(x) = x^{3} - 6 x^{2} + 9 x + 1$?

Explanation

1. 01

   $f'(x) = 3 \left(x - 3\right) \left(x - 1\right) = 0$ at $x = 1$ and $x = 3$.

2. 02

   $f''(x) = 6 x - 12$: $f''(1) = -6$, so $x = 1$ is a local maximum.

3. 03

   The value there is $f(1) = 5$.