https://file.aiquickdraw.com/mathbot/file/1727069032932bomiaz63.jpg

We are given the graph of \( f'(x) \), the derivative of \( f(x) \), and we need to determine:

1. **Where \( f(x) \) is increasing.**
2. **Where \( f(x) \) has an inflection point.**

### Understanding the graph:

- \( f'(x) \) is positive where the graph is above the x-axis, which means \( f(x) \) is increasing in those intervals.
- \( f'(x) \) is negative where the graph is below the x-axis, which means \( f(x) \) is decreasing in those intervals.
- Inflection points occur where \( f'(x) \) changes sign, which indicates a change in the concavity of \( f(x) \).

### First: The function \( f(x) \) is increasing on...
The function \( f(x) \) is increasing where \( f'(x) > 0 \), meaning the curve of \( f'(x) \) is above the x-axis.

From the graph:
- \( f'(x) > 0 \) on two intervals:
  - \( (2, 3) \) (between 2 and 3)
  - \( (4, \infty) \) (from 4 onwards)

Thus, the correct answer is:
- \( f(x) \) is increasing on both intervals \( (2, 3) \) and \( (4, \infty) \).
So, the answer is **(d) both (a) and (b)**.

### Second: The function \( f(x) \) has an inflection point at...
An inflection point occurs where \( f'(x) \) changes sign. This happens at points where \( f'(x) \) crosses the x-axis.

From the graph:
- \( f'(x) \) changes sign at \( x = 2 \), \( x = 3 \), and \( x = 4 \).

Thus, the function \( f(x) \) has inflection points at \( x = 2 \), \( x = 3 \), and \( x = 4 \).
So, the correct answer is **(d) All the previous**.

### Sketching \( f(x) \)
To sketch \( f(x) \), we can infer the following from \( f'(x) \):
- \( f(x) \) is **increasing** on \( (2, 3) \) and \( (4, \infty) \).
- \( f(x) \) is **decreasing** on \( (-\infty, 2) \) and \( (3, 4) \).
- \( f(x) \) has **inflection points** at \( x = 2 \), \( x = 3 \), and \( x = 4 \).

Based on this, the graph of \( f(x) \) will have:
- A decreasing slope from \( (-\infty, 2) \),
- A local minimum at \( x = 2 \),
- Increasing slope between \( (2, 3) \),
- A local maximum at \( x = 3 \),
- Decreasing slope between \( (3, 4) \),
- A local minimum at \( x = 4 \),
- Increasing slope for \( x > 4 \).

Do you want me to generate a sketch of \( f(x) \)?

### Follow-up questions:
1. What does it mean if the derivative \( f'(x) \) crosses the x-axis?
2. How does the sign of \( f'(x) \) affect the behavior of \( f(x) \)?
3. What would happen if \( f'(x) \) does not change sign at all?
4. How can we find concavity from the second derivative?
5. What are the implications of an inflection point on the graph of \( f(x) \)?

### Tip:
When analyzing the graph of \( f'(x) \), always focus on where it crosses the x-axis and its sign to understand the behavior of \( f(x) \).

The opposite figure represents the curve of the function f'(x), then:

First: The function f is increasing on:
(a) ]2, 3[ only
(b) ]4, ∞[ only
(c) ℝ
(d) both (a), (b)

Second: The function f has an inflection point at:
(a) x = 2
(b) x = 3
(c) x = 4
(d) All the previous.

Solve a calculus problem by analyzing the graph of f'(x) to determine where f(x) is increasing and where inflection points occur. This involves applying the first derivative test and understanding critical points. Suitable for students in Grades 10-12.

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