According to a scientists, he applied a force `F = -cx^(1//3)` on a particle and the particle is performing SHM. No other force acted on the particle. He refuses to tell whether `c` is a constant or not. Assume that he had worked only with positive `x` then :

Options

A. as `x` increases `c` also increases
B. as `x` increases `c` also decreases
C. as `x` increases `c` remain constant
D. the motion cannot be SHM

Answer

To solve the problem, we need to analyze the force applied to the particle and its implications for simple harmonic motion (SHM). The force given is \( F = -c x^{1/3} \).

### Step-by-Step Solution:

1. **Understanding the Force**:
The force acting on the particle is given by \( F = -c x^{1/3} \). This indicates that the force is dependent on the displacement \( x \) raised to the power of \( \frac{1}{3} \).

2. **Identifying the Form of SHM**:
In simple harmonic motion, the restoring force is typically proportional to the negative of the displacement, which can be expressed as \( F = -k x \), where \( k \) is a constant. For SHM to occur, the force must be linear in \( x \).

3. **Comparing with SHM Condition**:
The given force \( F = -c x^{1/3} \) does not have a linear relationship with \( x \) (since \( x^{1/3} \) is not linear). Therefore, this force does not satisfy the condition for SHM.

4. **Conclusion about the Nature of Motion**:
Since the force does not lead to a linear restoring force, the particle cannot be undergoing simple harmonic motion. Thus, the correct conclusion is that the motion cannot be simple harmonic motion.

5. **Evaluating the Options**:
Based on the analysis, we can evaluate the options provided:
- Option 1: As \( x \) increases, \( c \) also increases. (Not necessarily true)
- Option 2: As \( x \) increases, \( c \) also decreases. (Not necessarily true)
- Option 3: As \( x \) increases, \( c \) remains constant. (This is the most plausible if \( c \) is considered constant)
- Option 4: The motion cannot be simple harmonic motion. (This is correct)

Thus, the correct answer is that the motion cannot be simple harmonic motion.

### Final Answer:
The correct option is: **The motion cannot be simple harmonic motion.**