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To analyze this problem, let's go through the steps to find the critical points and determine the behavior of \( f(x) \) on the given intervals.

### Problem Overview
Given the function:
\[
f(x) = -2x^3 + 24x^2 - 72x + 1
\]
We are asked to:
1. Find the critical points \( A \) and \( B \).
2. Determine whether \( f(x) \) is increasing (INC) or decreasing (DEC) on the intervals \( (-\infty, A] \), \( [A, B] \), and \( [B, \infty) \).

### Solution Steps

#### Step 1: Find \( f'(x) \)
To find the critical points, we need the derivative of \( f(x) \):
\[
f(x) = -2x^3 + 24x^2 - 72x + 1
\]
Applying the power rule:
\[
f'(x) = -6x^2 + 48x - 72
\]

#### Step 2: Set \( f'(x) = 0 \) to Find Critical Points
We solve for \( x \) by setting \( f'(x) = 0 \):
\[
-6x^2 + 48x - 72 = 0
\]
Dividing by \(-6\) for simplicity:
\[
x^2 - 8x + 12 = 0
\]
Now we factor the quadratic:
\[
(x - 6)(x - 2) = 0
\]
Thus, \( x = 6 \) and \( x = 2 \) are the critical points. Let \( A = 2 \) and \( B = 6 \).

#### Step 3: Analyze \( f'(x) \) on Each Interval
To determine if \( f(x) \) is increasing or decreasing on each interval, we test the sign of \( f'(x) \) on the intervals \( (-\infty, A] \), \( [A, B] \), and \( [B, \infty) \):
1. **Interval \( (-\infty, 2) \)**: Choose a test point like \( x = 0 \):
   \[
   f'(0) = -6(0)^2 + 48(0) - 72 = -72
   \]
   Since \( f'(0) < 0 \), \( f(x) \) is **decreasing (DEC)** on \( (-\infty, 2) \).

2. **Interval \( (2, 6) \)**: Choose a test point like \( x = 4 \):
   \[
   f'(4) = -6(4)^2 + 48(4) - 72 = 24
   \]
   Since \( f'(4) > 0 \), \( f(x) \) is **increasing (INC)** on \( (2, 6) \).

3. **Interval \( (6, \infty) \)**: Choose a test point like \( x = 8 \):
   \[
   f'(8) = -6(8)^2 + 48(8) - 72 = -72
   \]
   Since \( f'(8) < 0 \), \( f(x) \) is **decreasing (DEC)** on \( (6, \infty) \).

### Summary
- **Critical Points**: \( A = 2 \), \( B = 6 \)
- **Intervals**:
  - \( (-\infty, 2) \): DEC
  - \( (2, 6) \): INC
  - \( (6, \infty) \): DEC

Would you like further details on each step or have any specific questions?

### Related Questions
1. How would we find inflection points for \( f(x) \) and their impact on concavity?
2. What are the second derivative tests, and how do they confirm the nature of critical points?
3. How does the behavior of the polynomial's degree affect the function's end behavior?
4. What if the function had more critical points? How would this change the analysis?
5. How do we handle rational functions when finding intervals of increase and decrease?

#### Tip
When analyzing polynomial functions, always simplify derivatives to their lowest terms to make factorization easier.

Consider the function f(x) = -2x^3 + 24x^2 - 72x + 1. For this function, there are three important intervals: (-∞, A], [A, B], and [B, ∞) where A and B are the critical points. Find A and B. For each of the following intervals, tell whether f(x) is increasing (type INC) or decreasing (type DEC).

Explore how to find the critical points and intervals of increase or decrease for the function f(x) = -2x^3 + 24x^2 - 72x + 1. This calculus problem walks through taking derivatives, setting the derivative to zero to find critical points, and using test points to determine where the function increases or decreases on specific intervals.

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