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1. What are the critical values of f(x)? 
2. On what intervals is f increasing? 
3. On what intervals is f decreasing? 
4. Find the x values of the local maxima 5. Find the x values of the local minima  

### Part (a): Critical values of \( f(x) \)

The graph provided is of \( f'(x) \), which represents the derivative of \( f(x) \). The critical values of \( f(x) \) occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined. From the graph, the derivative \( f'(x) \) crosses the x-axis (i.e., \( f'(x) = 0 \)) at the following points:

1. \( x = -4 \)
2. \( x = 0 \)
3. \( x = 3 \)

These are the critical points of \( f(x) \).

### Part (b): Intervals where \( f(x) \) is increasing

The function \( f(x) \) is increasing when \( f'(x) > 0 \), meaning the derivative is positive. Based on the graph, \( f'(x) > 0 \) on the following intervals:

- \( (-\infty, -4) \)
- \( (0, 3) \)
- \( (3, \infty) \)

### Part (c): Intervals where \( f(x) \) is decreasing

The function \( f(x) \) is decreasing when \( f'(x) < 0 \), meaning the derivative is negative. From the graph, \( f'(x) < 0 \) on the interval:

- \( (-4, 0) \)

### Part (d): Local maxima of \( f(x) \)

A local maximum occurs where \( f'(x) \) changes from positive to negative. From the graph, this occurs at:

- \( x = -4 \)

### Part (e): Local minima of \( f(x) \)

A local minimum occurs where \( f'(x) \) changes from negative to positive. From the graph, this happens at:

- \( x = 0 \)
- \( x = 3 \)

Would you like further details or clarifications on any part? Here are some follow-up questions to consider:

1. How can we use the second derivative to confirm the nature of critical points?
2. How do the concavity and inflection points of \( f(x) \) relate to \( f'(x) \)?
3. Can you determine the behavior of \( f(x) \) at the endpoints of the intervals?
4. What happens to \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \)?
5. How could we sketch the rough shape of \( f(x) \) based on this derivative graph?

**Tip:** Critical points are often just the start! Always examine the derivative's sign changes to understand the full behavior of the function.

1. What are the critical values of f(x)? 2. On what intervals is f increasing? 3. On what intervals is f decreasing? 4. Find the x values of the local maxima. 5. Find the x values of the local minima.

Analyze the derivative graph f'(x) to find critical values, determine where the function f is increasing or decreasing, and identify local maxima and minima. Suitable for high school calculus students (Grades 11-12).

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