Let’s work through the analysis of $f$ and its derivative $f'$ systematically based on the given graph and properties.

(a) Open intervals on which $f'(x) > 0$

$f'(x) > 0$ means the slope of $f$ is positive, which corresponds to $f$ increasing.

• From the description:
  • $f$ is increasing on $(0, 2)$ (positive slope between turning points at $x = 0$ and $x = 2$).
  • Therefore, $f'(x) > 0$ on $(0, 2)$.

Behavior of $f$:

• On $(0, 2)$, $f$ is increasing.

(b) Open intervals on which $f'(x) < 0$

$f'(x) < 0$ means the slope of $f$ is negative, which corresponds to $f$ decreasing.

• From the description:
  • $f$ is decreasing on $(-\infty, 0) \cup (2, \infty)$.
  • Therefore, $f'(x) < 0$ on $(-\infty, 0) \cup (2, \infty)$.

Behavior of $f$:

• On $(-\infty, 0) \cup (2, \infty)$, $f$ is decreasing.

(c) Open intervals on which $f'(x)$ is increasing

$f'(x)$ is increasing where the concavity of $f$ is upward (slopes of $f$ are becoming steeper in the positive or negative direction).

• From the description:
  • $f'(x)$ is increasing on $(-\infty, 1)$ because $f$ transitions from steep negative slope to steep positive slope (concave up).
  • Therefore, $f'(x)$ is increasing on $(-\infty, 1)$.

Behavior of $f$:

• On $(-\infty, 1)$, $f$ is concave upward.

(d) Open intervals on which $f'(x)$ is decreasing

$f'(x)$ is decreasing where the concavity of $f$ is downward (slopes of $f$ are becoming less steep in the positive or negative direction).

• From the description:
  • $f'(x)$ is decreasing on $(1, \infty)$ because $f$ transitions from steep positive slope to steep negative slope (concave down).
  • Therefore, $f'(x)$ is decreasing on $(1, \infty)$.

Behavior of $f$:

• On $(1, \infty)$, $f$ is concave downward.

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Final Answers:

(a) Intervals where $f'(x) > 0$:

• $(0, 2)$
• Behavior of $f$: $f$ is increasing.

(b) Intervals where $f'(x) < 0$:

• $(-\infty, 0) \cup (2, \infty)$
• Behavior of $f$: $f$ is decreasing.

(c) Intervals where $f'(x)$ is increasing:

• $(-\infty, 1)$
• Behavior of $f$: $f$ is concave upward.

(d) Intervals where $f'(x)$ is decreasing:

• $(1, \infty)$
• Behavior of $f$: $f$ is concave downward.

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Would you like a visual explanation or further clarification?
Here are 5 questions to expand on this:

1. What is the significance of $f''(x)$ in understanding $f'(x)$?
2. How does the second derivative relate to intervals of concavity for $f$?
3. Can you derive exact turning points using specific slope values from the graph?
4. How can you sketch $f'$ based on the features of $f$?
5. What happens at points of inflection where $f''(x) = 0$?

Tip: When sketching $f'$, look for changes in slope direction (zero crossings and maximum/minimum slopes) to pinpoint its shape.