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

Options

A.
B.
C.
D.

Answer

To determine which of the given graphs represents a function whose derivative has a maximum in the interval (0,1), we need to analyze the behavior of the derivatives of the functions represented by the graphs. Here is a step-by-step solution:

### Step 1: Understand the Concept of Derivatives
The derivative of a function gives us the slope of the tangent line at any point on the graph. If we denote the function as \( f(x) \), then its derivative is \( f'(x) \). A maximum of the derivative occurs when \( f'(x) \) reaches its highest value.

### Step 2: Identify the Condition for Maximum of Derivative
For a function \( g(x) = f'(x) \) to have a maximum at a point \( x_0 \) in the interval (0,1), the following conditions must be satisfied:
1. \( g'(x_0) = 0 \) (the first derivative of \( g \) is zero).
2. \( g''(x_0) < 0 \) (the second derivative of \( g \) is negative, indicating a maximum).

### Step 3: Analyze the Given Graphs
We need to look at the graphs provided and determine where the derivative \( g(x) \) has a maximum. We will check for points in the interval (0,1) where the slope of the tangent (the derivative) is at its peak.

- **Graph A**: The curve appears to have a peak in the interval (0,1). The slope increases to a maximum and then decreases, indicating that the derivative has a maximum in this interval.

- **Graph B**: Similar to Graph A, this graph shows a peak in the interval (0,1), suggesting that the derivative also has a maximum here.

- **Graph C**: This graph does not show a change in concavity and does not have a peak in the interval (0,1). Therefore, the derivative does not have a maximum here.

- **Graph D**: This graph also shows a peak in the interval (0,1), indicating that the derivative has a maximum.

### Step 4: Conclusion
From the analysis:
- Graphs A, B, and D represent functions whose derivatives have a maximum in the interval (0,1).
- Graph C does not represent such a function.

### Final Answer
The graphs that represent functions whose derivatives have a maximum in the interval (0,1) are **A, B, and D**.