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Let's work through the problem.

We are given the derivative of a function, \( f'(x) = 3x(x + 5) \), and asked to answer the following questions:

### a. What are the critical points of \( f \)?
Critical points occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined. Here, \( f'(x) = 3x(x + 5) \) is a product of two factors, and it's a polynomial, so it is defined everywhere. We set \( f'(x) = 0 \) to find the critical points:
\[
3x(x + 5) = 0
\]
This gives two solutions:
\[
x = 0 \quad \text{or} \quad x = -5
\]
So, the critical points are \( x = 0 \) and \( x = -5 \).

### b. On what open intervals is \( f \) increasing or decreasing?
To determine where the function is increasing or decreasing, we look at the sign of \( f'(x) = 3x(x + 5) \).

- **When is \( f'(x) > 0 \)?** This happens when the product \( 3x(x + 5) \) is positive, which occurs when both factors are either positive or both are negative.
    - For \( x > 0 \), both \( x > 0 \) and \( x + 5 > 0 \), so \( f'(x) > 0 \) (increasing).
    - For \( -5 < x < 0 \), \( x < 0 \) and \( x + 5 > 0 \), so \( f'(x) < 0 \) (decreasing).
    - For \( x < -5 \), both \( x < 0 \) and \( x + 5 < 0 \), so \( f'(x) > 0 \) (increasing).

Therefore, \( f \) is increasing on \( (-\infty, -5) \cup (0, \infty) \) and decreasing on \( (-5, 0) \).

### c. At what points, if any, does \( f \) assume local maximum and minimum values?
To find local extrema, we check the critical points \( x = -5 \) and \( x = 0 \), and determine the behavior of \( f'(x) \) around these points:

- At \( x = -5 \), the function changes from increasing (left of \( -5 \)) to decreasing (right of \( -5 \)). Therefore, \( f \) has a **local maximum** at \( x = -5 \).
- At \( x = 0 \), the function changes from decreasing (left of \( 0 \)) to increasing (right of \( 0 \)). Therefore, \( f \) has a **local minimum** at \( x = 0 \).

### Summary:
- **Critical points**: \( x = -5 \) and \( x = 0 \)
- **Increasing**: \( (-\infty, -5) \cup (0, \infty) \)
- **Decreasing**: \( (-5, 0) \)
- **Local maximum**: At \( x = -5 \)
- **Local minimum**: At \( x = 0 \)

Would you like more details on any part of the solution?

Here are 5 related questions:
1. How do you find inflection points of a function?
2. Can a function have a critical point that is not a local maximum or minimum?
3. What is the second derivative test, and how does it work for finding concavity?
4. How do we sketch the graph of \( f(x) \) from its derivative?
5. What happens if the derivative has higher-order roots at critical points?

**Tip**: Remember, critical points occur not just where the derivative equals zero but also where it's undefined! Always check for both.

Answer the following questions about the function whose derivative is f'(x) = 3x(x + 5). a. What are the critical points of f? b. On what open intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values?

This problem explores how to find the critical points, intervals of increase and decrease, and local extrema of a function given its derivative f'(x) = 3x(x + 5). Step-by-step explanations help solve each part, making it suitable for students in Grades 11-12 studying calculus.

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