x^3-6x^2+8 [1,6] intervals in which f is increasing or decreasing

To determine where the function \( f(x) = x^3 - 6x^2 + 8 \) is increasing or decreasing over the interval \( [1, 6] \), we need to:

1. Find the derivative \( f'(x) \) to identify critical points.
2. Determine where \( f'(x) \) is positive or negative.

### Step 1: Calculate \( f'(x) \)
The derivative of \( f(x) \) is:
\[
f'(x) = 3x^2 - 12x
\]
This simplifies to:
\[
f'(x) = 3x(x - 4)
\]

### Step 2: Find Critical Points
Set \( f'(x) = 0 \

Explore how to find the intervals where f(x) = x^3 - 6x^2 + 8 is increasing or decreasing over the interval [1, 6]. This calculus problem involves differentiation, finding critical points, and applying the first derivative test, making it ideal for high school students in Grades 11-12.

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