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Let's analyze the graph of \( f'(x) \), which is the derivative of \( f(x) \), to answer the questions provided:

1. **(a) Critical Number(s):**
   - Critical numbers occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined. From the graph, \( f'(x) = 0 \) at \( x = 0 \), because the derivative touches the x-axis at this point.

   **Answer: \( x = 0 \)**

2. **(b) Increasing Interval(s):**
   - \( f(x) \) is increasing where \( f'(x) > 0 \) (above the x-axis).
   - Based on the graph, \( f'(x) > 0 \) on the intervals \( (0, \infty) \).

   **Answer: \( (0, \infty) \)**

3. **(c) Decreasing Interval(s):**
   - \( f(x) \) is decreasing where \( f'(x) < 0 \) (below the x-axis).
   - From the graph, \( f'(x) < 0 \) on the interval \( (-\infty, 0) \).

   **Answer: \( (-\infty, 0) \)**

4. **(d) Relative Maxima Occur at \( x = \):**
   - Relative maxima occur where \( f'(x) \) changes from positive to negative. This is not the case anywhere on this graph, as there is no transition from positive to negative slope.

   **Answer: NO SOLUTION**

5. **(e) Relative Minima Occur at \( x = \):**
   - Relative minima occur where \( f'(x) \) changes from negative to positive, which happens at \( x = 0 \).

   **Answer: \( x = 0 \)**

6. **(f) Concave Up Interval(s):**
   - \( f(x) \) is concave up where \( f'(x) \) is increasing, which corresponds to the interval where the slope of \( f'(x) \) is positive.
   - From the graph, \( f'(x) \) is increasing on the interval \( (0, \infty) \).

   **Answer: \( (0, \infty) \)**

7. **(g) Concave Down Interval(s):**
   - \( f(x) \) is concave down where \( f'(x) \) is decreasing, which corresponds to the interval where the slope of \( f'(x) \) is negative.
   - From the graph, \( f'(x) \) is decreasing on the interval \( (-\infty, 0) \).

   **Answer: \( (-\infty, 0) \)**

8. **(h) Inflection Points Occur at \( x = \):**
   - Inflection points occur where the concavity of \( f(x) \) changes, meaning where \( f'(x) \) changes from increasing to decreasing or vice versa. This happens at \( x = 0 \), where the slope of \( f'(x) \) changes.

   **Answer: \( x = 0 \)**

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Would you like more details or have any questions on any of these parts? 

Here are 5 follow-up questions that may help expand on this information:
1. How do critical points affect the shape of the graph of \( f(x) \)?
2. Why do concave up intervals indicate increasing concavity in \( f(x) \)?
3. What is the significance of inflection points in the graph of a function?
4. Can a point of relative minima or maxima be classified as an inflection point as well?
5. What is the relationship between the first and second derivative in identifying concavity?

**Tip**: When analyzing the graph of the derivative, always keep in mind that positive values of \( f'(x) \) indicate increasing intervals, while negative values indicate decreasing intervals for \( f(x) \).

Consider the graph of f'(x), which represents the derivative of f(x). Analyze the graph to find critical points, intervals of increase and decrease, relative maxima and minima, concavity, and inflection points.

Analyze the graph of f'(x) to determine critical points, intervals of increase and decrease, relative extrema, concavity, and inflection points. Suitable for high school students studying calculus.

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