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To solve this problem, let’s analyze the derivative \( g'(x) = x(x + 8)(x + 1)^2(x - 5) \).

### Part (a): Determine intervals of increase and decrease
To find where \( g(x) \) is increasing or decreasing, we need to identify the critical points by setting \( g'(x) = 0 \) and analyzing the sign changes of \( g'(x) \).

1. **Find Critical Points**:
   \[
   g'(x) = x(x + 8)(x + 1)^2(x - 5) = 0
   \]
   The solutions are:
   - \( x = 0 \)
   - \( x = -8 \)
   - \( x = -1 \) (note that this factor is squared, which will affect the sign analysis)
   - \( x = 5 \)

2. **Sign Analysis**:
   Divide the number line into intervals based on these critical points: \( (-\infty, -8) \), \( (-8, -1) \), \( (-1, 0) \), \( (0, 5) \), and \( (5, \infty) \).

   We’ll test each interval to determine the sign of \( g'(x) \):
   - For \( x < -8 \): Choose a test point (e.g., \( x = -9 \)). The product \( g'(-9) \) will be positive.
   - For \( -8 < x < -1 \): Choose \( x = -5 \). The product \( g'(-5) \) will be negative.
   - For \( -1 < x < 0 \): Choose \( x = -0.5 \). The product \( g'(-0.5) \) will be negative.
   - For \( 0 < x < 5 \): Choose \( x = 1 \). The product \( g'(1) \) will be positive.
   - For \( x > 5 \): Choose \( x = 6 \). The product \( g'(6) \) will be positive.

3. **Intervals**:
   - \( g(x) \) is **increasing** on \( (-\infty, -8) \), \( (0, 5) \), and \( (5, \infty) \).
   - \( g(x) \) is **decreasing** on \( (-8, -1) \) and \( (-1, 0) \).

### Part (b): Determine the locations and types of local extrema
Using the information from the intervals:

- **Relative Minima** occur where \( g(x) \) changes from decreasing to increasing:
  - At \( x = -8 \)
  - At \( x = 5 \)

- **Relative Maxima** occur where \( g(x) \) changes from increasing to decreasing:
  - At \( x = 0 \)

### Final Answers
#### Part (a)
- \( g(x) \) is decreasing on \( (-8, -1) \) and \( (-1, 0) \).
- \( g(x) \) is increasing on \( (-\infty, -8) \), \( (0, 5) \), and \( (5, \infty) \).

#### Part (b)
- \( g(x) \) has relative minima at \( x = -8, 5 \).
- \( g(x) \) has a relative maximum at \( x = 0 \).

Would you like further details on any steps?

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### Follow-up Questions
1. How would the answer change if one of the roots was a triple root instead of a double?
2. What would the graph of \( g(x) \) look like based on these intervals?
3. How can we determine concavity from the second derivative of \( g(x) \)?
4. How would the intervals change if we had additional factors in \( g'(x) \)?
5. Can we use this derivative test for non-polynomial functions as well?

### Tip
When analyzing polynomial derivatives, pay close attention to multiplicities of factors (like squares or cubes) because they affect the behavior at critical points.

Given that g is continuous everywhere and has derivative: g'(x) = x(x + 8)(x + 1)^2(x - 5), determine the intervals of increase and decrease of g, and identify any local extrema.

Learn how to analyze the function g(x) based on its derivative g'(x) = x(x + 8)(x + 1)^2(x - 5). Determine intervals of increase and decrease, and find the locations and types of local extrema using calculus. Suitable for students in Grade 11-12 or college-level calculus.

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