https://file.aiquickdraw.com/mathbot/file/17286554866584304rp9c.jpg

The image contains a mathematical problem involving a polynomial function and its intervals of increase or decrease, as well as a graph. To assist, I will extract and solve the problem step by step.

The given polynomial function is:

\[
f(x) = x(x - 3)(x + 2)
\]

The question asks for the intervals on which this function is increasing or decreasing. Let’s go through the solution process for this.

### Steps:
1. **Find the derivative of \( f(x) \)** to determine the critical points.
2. **Analyze the sign of the derivative** in different intervals to see where the function is increasing or decreasing.
3. **Confirm the behavior using the graph** provided.

I will now solve this.The derivative of the function \( f(x) = x(x - 3)(x + 2) \) is:

\[
f'(x) = x(x - 3) + x(x + 2) + (x - 3)(x + 2)
\]

Solving \( f'(x) = 0 \) gives two critical points:

\[
x = \frac{1}{3} - \frac{\sqrt{19}}{3} \quad \text{and} \quad x = \frac{1}{3} + \frac{\sqrt{19}}{3}
\]

These are the points where the function changes its increasing or decreasing behavior.

Now, let's analyze the intervals around these critical points to determine where the function is increasing and decreasing.Based on the analysis, the function behaves as follows:

- **Increasing** on the interval \( (-\infty, -1.12) \)
- **Decreasing** on the interval \( (-1.12, 1.79) \)
- **Increasing** again on the interval \( (1.79, \infty) \)

Thus, the function is increasing and decreasing over these intervals.

Would you like further details, or to see how this relates to the graph provided?

Here are 5 related questions for practice:
1. How do you find the critical points of a polynomial function?
2. What does the sign of the derivative tell you about the behavior of the function?
3. How do you determine where a function is increasing or decreasing?
4. What is the importance of analyzing the second derivative in such problems?
5. How do inflection points relate to the graph of a polynomial?

**Tip:** Always check the sign of the derivative between critical points to determine the function's behavior (increase/decrease).

The polynomial function f is given by f(x) = x(x - 3)(x + 2). What are the intervals on which f(x) is increasing or decreasing?

Learn how to find the intervals of increase and decrease for the polynomial function f(x) = x(x - 3)(x + 2) by analyzing its derivative and critical points. This detailed guide uses the first derivative test to determine the behavior of the function across different intervals.

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