Which of the following graphs represent function whose derivatives have a maximum in the interval `(0,1)` ?

Options

A. B
B.
C.
D.

Answer

To determine which of the given graphs represents a function whose derivatives have a maximum in the interval (0, 1), we need to analyze the concavity of the function and its derivative.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find a function \( f(x) \) such that its derivative \( f'(x) \) has a maximum in the interval \( (0, 1) \).
- This means that \( f'(x) \) should change its concavity in this interval.

2. **Concept of Concavity**:
- If \( f'(x) \) has a maximum in the interval \( (0, 1) \), then the second derivative \( f''(x) \) must change sign in this interval.
- Specifically, \( f''(x) \) should be positive before the maximum and negative after the maximum, indicating a change from concave up to concave down.

3. **Analyzing the Options**:
- We will examine each option to see if the graph of the function \( f(x) \) shows a change in concavity in the interval \( (0, 1) \).

4. **Option 1**:
- If the graph is concave up before \( x = 1 \) and concave down after \( x = 1 \), then \( f'(x) \) has a maximum in the interval \( (0, 1) \).
- This option satisfies the condition.

5. **Option 2**:
- Similar to Option 1, if the graph shows a change from concave up to concave down in the interval \( (0, 1) \), then this option also satisfies the condition.

6. **Option 3**:
- If the graph does not show a change in concavity or remains either concave up or concave down throughout the interval, then this option does not satisfy the condition.

7. **Option 4**:
- If the graph shows a change from concave up to concave down in the interval \( (0, 1) \), then this option satisfies the condition.

8. **Conclusion**:
- Based on the analysis, Options 1, 2, and 4 represent functions whose derivatives have a maximum in the interval \( (0, 1) \).
- Therefore, the final answer is: **Options 1, 2, and 4 are correct**.