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To solve this problem, let's start by finding the critical points \( A \) and \( B \) for the function \( f(x) = -2x^3 + 27x^2 - 48x + 1 \).

### Step 1: Find the derivative of \( f(x) \)
To find the critical points, we need to take the derivative of \( f(x) \) and set it equal to zero.

\[
f(x) = -2x^3 + 27x^2 - 48x + 1
\]
\[
f'(x) = \frac{d}{dx}(-2x^3 + 27x^2 - 48x + 1) = -6x^2 + 54x - 48
\]

### Step 2: Solve \( f'(x) = 0 \)
Set the derivative equal to zero to find the values of \( x \) that make \( f'(x) = 0 \).

\[
-6x^2 + 54x - 48 = 0
\]

Divide by \(-6\) to simplify:

\[
x^2 - 9x + 8 = 0
\]

Now, factor the quadratic equation:

\[
(x - 1)(x - 8) = 0
\]

This gives us two solutions:

\[
x = 1 \quad \text{and} \quad x = 8
\]

So, the critical points are \( A = 1 \) and \( B = 8 \).

### Step 3: Determine the behavior of \( f(x) \) in each interval
We need to check if \( f(x) \) is increasing or decreasing in each of the intervals \( (-\infty, A] \), \( [A, B] \), and \( [B, \infty) \). To do this, we can use the sign of \( f'(x) \) in each interval.

1. **Interval \( (-\infty, 1) \)**: Choose a test point, such as \( x = 0 \):
   - \( f'(0) = -6(0)^2 + 54(0) - 48 = -48 \) (negative), so \( f(x) \) is **decreasing (DEC)** in this interval.

2. **Interval \( (1, 8) \)**: Choose a test point, such as \( x = 5 \):
   - \( f'(5) = -6(5)^2 + 54(5) - 48 = -150 + 270 - 48 = 72 \) (positive), so \( f(x) \) is **increasing (INC)** in this interval.

3. **Interval \( (8, \infty) \)**: Choose a test point, such as \( x = 10 \):
   - \( f'(10) = -6(10)^2 + 54(10) - 48 = -600 + 540 - 48 = -108 \) (negative), so \( f(x) \) is **decreasing (DEC)** in this interval.

### Summary of Results
- \( A = 1 \), \( B = 8 \)
- \( (-\infty, A] \): **DEC**
- \( [A, B] \): **INC**
- \( [B, \infty) \): **DEC**

Would you like more details on any step, or have any questions?

### Related Questions
1. How would you verify whether \( f(x) \) has any local maxima or minima at points \( A \) and \( B \)?
2. How would the intervals change if the function \( f(x) \) had an additional critical point?
3. How do we interpret the behavior of \( f(x) \) based on the sign of \( f'(x) \)?
4. Could there be any inflection points for this function, and how would you find them?
5. What are the potential applications of finding increasing and decreasing intervals in real-world scenarios?

### Tip
Remember, the first derivative test is a useful tool for identifying local maxima and minima and understanding the function's behavior on different intervals.

Consider the function f(x) = -2x^3 + 27x^2 - 48x + 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 in INC) or decreasing (type in DEC).

Learn how to find critical points for the function f(x) = -2x^3 + 27x^2 - 48x + 1 and analyze the increasing or decreasing behavior over specified intervals. This calculus problem covers derivatives, critical points, and the First Derivative Test, ideal for Grades 11-12.

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