Answer:
To find the displacement when the velocity is zero, you need to determine the value of \( x \) when \( \frac{dx}{dt} = 0 \) (velocity is zero). In your case, \( \frac{dx}{dt} \) is the derivative of the displacement equation \( \sqrt{3x} + 6 \) with respect to time.
Let's find the derivative:
\[ \frac{d}{dt}(\sqrt{3x} + 6) \]
The derivative of \( \sqrt{3x} \) with respect to \( x \) is \( \frac{1}{2\sqrt{3x}} \), and then you multiply it by the derivative of \( 3x \) with respect to \( t \) (which is 3). So, the derivative is:
\[ \frac{1}{2\sqrt{3x}} \cdot 3 \]
Now, set this equal to zero and solve for \( x \):
\[ \frac{1}{2\sqrt{3x}} \cdot 3 = 0 \]
Since the term \( \frac{1}{2\sqrt{3x}} \) cannot be zero, the only way for the whole expression to be zero is if the constant term (3) is zero. However, this is not possible. Therefore, there is no solution for \( x \) when the velocity is zero in this context, meaning the particle never comes to rest.
If you have additional information or constraints, please provide them for a more accurate analysis.