For a particle moving along `x-`axis, speed must be increasing for the following graph `:`

Options

A.
B.
C.
D.

Explanation

To determine when the speed of a particle moving along the x-axis is increasing, we need to analyze the relationship between velocity and acceleration. The speed of a particle increases when the velocity and acceleration have the same sign, meaning both are either positive or negative.

### Step-by-Step Solution:

1. **Understanding Speed and Velocity**:
- Speed is the magnitude of velocity. For speed to increase, the particle must be moving in the same direction as its acceleration.

2. **Identifying Conditions for Increasing Speed**:
- Case 1: If the velocity is positive and the acceleration is also positive, the speed is increasing.
- Case 2: If the velocity is negative and the acceleration is also negative, the speed is also increasing.

3. **Analyzing the Graph**:
- Look at the graph provided. The slope of the graph represents the velocity of the particle.
- A positive slope indicates positive velocity, while a negative slope indicates negative velocity.

4. **Determining the Slope**:
- Identify sections of the graph where the slope is positive (indicating positive velocity) and where the slope is negative (indicating negative velocity).
- Check the acceleration (the slope of the velocity graph). If the slope is increasing (positive), the acceleration is positive.

5. **Conclusion**:
- The speed of the particle is increasing when both the velocity and acceleration are either positive or negative.
- From the analysis, if the graph shows sections where both the velocity and acceleration are positive or both are negative, then the speed is increasing.

### Final Answer:
The correct answer is D, where the conditions for increasing speed are satisfied.