What is the greatest value of $f(x) = x^{3} - 6 x^{2} + 9 x + 2$ on $[0, 5]$?

Explanation

1. 01

   Critical points inside the interval: $f'(x) = 3 \left(x - 3\right) \left(x - 1\right) = 0$ at $x = 1, 3$.

2. 02

   Compare $f$ there and at the two ends: $f(0) = 2$, $f(5) = 22$, $f(1) = 6$, $f(3) = 2$.

3. 03

   The greatest value is $22$, at an end of the interval.