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To analyze the graph of the function \( f(x) \) and answer the questions about its derivative \( f'(x) \), let's follow these steps:

1. **Identify where \( f(x) \) has critical points** (i.e., where the derivative \( f'(x) \) is zero, indicating potential maxima, minima, or inflection points).
2. **Determine where \( f(x) \) is increasing or decreasing**, which will tell us where \( f'(x) \) is positive or negative.

### Analysis:

- **Critical Points**: These are points where \( f'(x) = 0 \). From the graph, critical points occur where the slope of the tangent is zero. This typically happens at the local maximum, minimum, or horizontal inflection points.
  - The graph appears to have a local minimum around \( x = -1 \).
  - The graph is neither steeply rising nor falling at this point, implying \( f'(x) = 0 \) here.

- **Increasing and Decreasing Intervals**:
  - **Decreasing**: From \( x = -7 \) to \( x \approx -1 \), \( f(x) \) is decreasing. Therefore, \( f'(x) \) is negative in this interval.
  - **Increasing**: From \( x \approx -1 \) to \( x = 7 \), \( f(x) \) is increasing. Therefore, \( f'(x) \) is positive in this interval.

### Answers:

a) **When does \( f'(x) \) cross the \( x \)-axis?**
   - \( f'(x) \) crosses the \( x \)-axis at the critical point where \( f(x) \) has a local minimum.
   - This occurs at \( x \approx -1 \).

b) **When is \( f'(x) \geq 0 \)?**
   - \( f'(x) \) is zero at \( x \approx -1 \).
   - \( f'(x) \) is positive for \( x > -1 \).
   - Thus, \( f'(x) \geq 0 \) for \( x \in [-1, 7] \).

c) **When is \( f'(x) \leq 0 \)?**
   - \( f'(x) \) is zero at \( x \approx -1 \).
   - \( f'(x) \) is negative for \( x < -1 \).
   - Thus, \( f'(x) \leq 0 \) for \( x \in [-7, -1] \).

### Summary:

- **a)** \( x \approx -1 \)
- **b)** \( [-1, 7] \)
- **c)** \( [-7, -1] \)

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:
1. How would you determine the concavity of \( f(x) \) using the second derivative \( f''(x) \)?
2. What is the significance of the inflection points on the graph of \( f(x) \)?
3. How do the intervals where \( f'(x) > 0 \) and \( f'(x) < 0 \) relate to the graph of \( f(x) \)?
4. How can you find the exact values of the critical points analytically if the function \( f(x) \) is given explicitly?
5. What are the implications of \( f'(x) \) being zero over an interval rather than just at a point?

**Tip**: Understanding the relationship between a function and its derivative is crucial for analyzing the behavior of the function, including identifying local maxima, minima, and points of inflection.

When does f'(x) cross the x-axis, and when is f'(x) greater than or equal to 0 or less than or equal to 0?

This problem involves identifying where f'(x) crosses the x-axis and determining intervals where f'(x) is positive or negative based on a graph of the function f(x). Suitable for high school students.

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