To determine whether there are local minima or maxima in the function $f(x) = x^3 - 6x^2 + 8$ on the interval $[1, 6]$, we will follow these steps:

1. Find the first derivative $f'(x)$ and set it to zero to locate critical points.
2. Find the second derivative $f''(x)$ to classify each critical point as a minimum, maximum, or neither.
3. Evaluate the function at each critical point and at the endpoints of the interval to determine the minimum and maximum values within $[1, 6]$.

Step 1: First Derivative

The first derivative of $f(x) = x^3 - 6x^2 + 8$ is: $f'(x) = 3x^2 - 12x$ We can factor this as: $f'(x) = 3x(x - 4)$ Setting $f'(x) = 0$: $3x(x - 4) = 0$ This gives $x = 0$ and $x = 4$ as critical points. However, since we are only interested in the interval ([