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: $x = 0$

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 $f(x)$.