https://file.aiquickdraw.com/mathbot/file/17256053108502d3y261n.jpg

The image shows a graph of a function \( f \) with a multiple-choice question asking which of the following statements must be true regarding the graph of the derivative \( f'(x) \):

The answer options are as follows:

A. \( f \) has a relative maximum at \( x = 1 \).
B. \( f \) has a relative minimum at \( x = 2 \).
C. \( f'(x) < 0 \) for \( 0 < x < 1 \).
D. \( f'(x) > 0 \) for \( 1 < x < 3 \).

Let's analyze the graph and options:
1. Option A suggests a relative maximum at \( x = 1 \). This would mean \( f'(1) = 0 \) and the slope changes from positive to negative.
2. Option B suggests a relative minimum at \( x = 2 \), meaning \( f'(2) = 0 \) and the slope changes from negative to positive.
3. Option C suggests that the slope of the function (or \( f'(x) \)) is negative between \( x = 0 \) and \( x = 1 \).
4. Option D suggests that the slope of the function is positive between \( x = 1 \) and \( x = 3 \).

I will now evaluate the graph and confirm which options are correct.Based on the graph and the problem's analysis:

1. **Option A: \( f \) has a relative maximum at \( x = 1 \).**
   - At \( x = 1 \), the function appears to reach a peak, meaning that \( f'(x) \) would change from positive to negative. This implies that there is a relative maximum at \( x = 1 \), so **Option A is true**.

2. **Option B: \( f \) has a relative minimum at \( x = 2 \).**
   - At \( x = 2 \), the function seems to have a valley, where \( f'(x) \) would change from negative to positive. This suggests that there is a relative minimum at \( x = 2 \), so **Option B is true**.

3. **Option C: \( f'(x) < 0 \) for \( 0 < x < 1 \).**
   - For values between \( x = 0 \) and \( x = 1 \), the function's slope is positive as the curve is increasing. Therefore, \( f'(x) > 0 \) in this interval, making **Option C false**.

4. **Option D: \( f'(x) > 0 \) for \( 1 < x < 3 \).**
   - For \( x \) between 1 and 3, the function first decreases and then starts increasing after \( x = 2 \). This means that the slope is not entirely positive in this interval, so **Option D is false**.

### Conclusion:
- The correct statements are **Option A** and **Option B**.

Would you like any further details or clarifications?

Here are five related questions to explore further:
1. How do you determine the concavity of a function from its graph?
2. What is the relationship between the first and second derivatives of a function?
3. How do inflection points relate to the second derivative?
4. What is the significance of critical points in finding maxima and minima?
5. How can you estimate the value of a derivative from a graph?

**Tip:** To determine where a function is increasing or decreasing, observe the slope of the tangent lines: increasing when positive, decreasing when negative.

The graph of a function f is given. Which of the following statements must be true about its derivative f'(x)? (A) f has a relative maximum at x = 1. (B) f has a relative minimum at x = 2. (C) f'(x) < 0 for 0 < x < 1. (D) f'(x) > 0 for 1 < x < 3.

Analyze the graph of a function to determine where its derivative is zero and whether the function has maxima or minima. Explore the first derivative test and examine increasing and decreasing intervals, suitable for students in Grades 11-12.

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