To solve the problem of determining the intervals where the function [tex]\( f(x) = x^3 - x^2 - x \)[/tex] is increasing, decreasing, or constant, we will follow these steps:
### Step 1: Find the First Derivative
First, we need to find the first derivative of the function, [tex]\( f'(x) \)[/tex], because the first derivative tells us about the slope of the function at any given point.
[tex]\[
f(x) = x^3 - x^2 - x
\][/tex]
[tex]\[
f'(x) = \frac{d}{dx}(x^3 - x^2 - x) = 3x^2 - 2x - 1
\][/tex]
### Step 2: Find the Critical Points
Next, we find the critical points by setting the first derivative equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[
3x^2 - 2x - 1 = 0
\][/tex]
Solving this quadratic equation, we get two critical points:
[tex]\[
x = 1 \quad \text{and} \quad x = -\frac{1}{3}
\][/tex]
### Step 3: Determine the Behavior on the Intervals
To determine where the function is increasing or decreasing, we need to test the intervals determined by the critical points.
The critical points split the number line into three intervals:
1. [tex]\( (-\infty, -\frac{1}{3}) \)[/tex]
2. [tex]\( (-\frac{1}{3}, 1) \)[/tex]
3. [tex]\( (1, \infty) \)[/tex]
We will pick test points in each interval to determine whether the function is increasing or decreasing in that interval.
- For [tex]\( (-\infty, -\frac{1}{3}) \)[/tex], choose [tex]\( x = -2 \)[/tex]:
[tex]\[
f'(-2) = 3(-2)^2 - 2(-2) - 1 = 12 + 4 - 1 = 15 \quad (\text{positive, so } f(x) \text{ is increasing})
\][/tex]
- For [tex]\( (-\frac{1}{3}, 1) \)[/tex], choose [tex]\( x = 0 \)[/tex]:
[tex]\[
f'(0) = 3(0)^2 - 2(0) - 1 = -1 \quad (\text{negative, so } f(x) \text{ is decreasing})
\][/tex]
- For [tex]\( (1, \infty) \)[/tex], choose [tex]\( x = 2 \)[/tex]:
[tex]\[
f'(2) = 3(2)^2 - 2(2) - 1 = 12 - 4 - 1 = 7 \quad (\text{positive, so } f(x) \text{ is increasing})
\][/tex]
### Step 4: Summarize the Intervals
From the above analysis, we can see:
- The function is increasing in the intervals [tex]\( (-\infty, -\frac{1}{3}) \)[/tex] and [tex]\( (1, \infty) \)[/tex].
- The function is decreasing in the interval [tex]\( (-\frac{1}{3}, 1) \)[/tex].
Therefore, the correct solution is:
The function [tex]\( f(x) = x^3 - x^2 - x \)[/tex] is decreasing from [tex]\(-\frac{1}{3}\)[/tex] to 1, and increasing otherwise.
