(a) The position of a particle moving along the x-axis depends on the time according to equation. `x=(3t^(2)-t^(3))` m. At what time does the particle reaches its maximum x-position?
(b) What is the displacement during the first 4s.

Explanation

To solve the given problem step by step, we will break it down into two parts as specified in the question.

### Part (a): Finding the time at which the particle reaches its maximum x-position

1. **Given Equation**: The position of the particle is given by the equation:
\[ x(t) = 3t^2 - t^3 \]

2. **Finding the Velocity**: To find the time when the particle reaches its maximum position, we need to find the velocity of the particle, which is the derivative of the position with respect to time:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(3t^2 - t^3) \]

3. **Calculating the Derivative**:
\[ v(t) = 6t - 3t^2 \]

4. **Setting the Velocity to Zero**: To find the maximum position, we set the velocity equal to zero:
\[ 6t - 3t^2 = 0 \]

5. **Factoring the Equation**:
\[ 3t(2 - t) = 0 \]

6. **Finding the Roots**: This gives us two solutions:
\[ t = 0 \quad \text{or} \quad t = 2 \]

7. **Conclusion for Part (a)**: The particle reaches its maximum x-position at \( t = 2 \) seconds.

### Part (b): Finding the displacement during the first 4 seconds

1. **Displacement Calculation**: Displacement is defined as the change in position from the initial time to the final time. We need to find the position at \( t = 0 \) seconds and \( t = 4 \) seconds.

2. **Finding Position at \( t = 0 \)**:
\[ x(0) = 3(0)^2 - (0)^3 = 0 \, \text{m} \]

3. **Finding Position at \( t = 4 \)**:
\[ x(4) = 3(4)^2 - (4)^3 \]
\[ = 3(16) - 64 \]
\[ = 48 - 64 = -16 \, \text{m} \]

4. **Calculating Displacement**: The displacement during the first 4 seconds is:
\[ \text{Displacement} = x(4) - x(0) = -16 - 0 = -16 \, \text{m} \]

5. **Conclusion for Part (b)**: The displacement during the first 4 seconds is \(-16\) meters.

### Summary of the Solution:
- (a) The particle reaches its maximum x-position at \( t = 2 \) seconds.
- (b) The displacement during the first 4 seconds is \(-16\) meters.