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.