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The problem asks about a particle moving along the x-axis, with its position given by the equation:

\[
x(t) = t^3 - 3t^2 + 9t - 1
\]

The question is to determine when the particle is at rest, which means the velocity of the particle is zero. The velocity is the first derivative of the position function with respect to time \( t \).

Let's calculate the velocity function \( v(t) = \frac{dx(t)}{dt} \) and find the values of \( t \) when the particle is at rest (i.e., when \( v(t) = 0 \)).

### Step 1: Calculate the velocity function

Given the position function:
\[
x(t) = t^3 - 3t^2 + 9t - 1
\]

The velocity function is the derivative of the position with respect to \( t \):
\[
v(t) = \frac{d}{dt}(t^3 - 3t^2 + 9t - 1)
\]
\[
v(t) = 3t^2 - 6t + 9
\]

### Step 2: Set velocity equal to zero to find when the particle is at rest

To find when the particle is at rest, we solve for \( t \) when \( v(t) = 0 \):
\[
3t^2 - 6t + 9 = 0
\]

### Step 3: Solve the quadratic equation

Dividing the equation by 3 to simplify:
\[
t^2 - 2t + 3 = 0
\]

Now, solve the quadratic equation using the discriminant method:
\[
\Delta = (-2)^2 - 4 \times 1 \times 3 = 4 - 12 = -8
\]

Since the discriminant \( \Delta = -8 \) is negative, there are no real solutions to the equation, meaning the velocity never equals zero.

### Conclusion

Since there are no real values for \( t \) when \( v(t) = 0 \), the particle is never at rest.

The correct answer is **(A) No values**.

Would you like further details or assistance with another question? 

Here are some related questions:
1. How do you interpret the meaning of negative discriminants in physics?
2. What other physical concepts are related to velocity and acceleration?
3. How would the graph of the position function \( x(t) \) look?
4. What happens if the particle's velocity function has a positive discriminant?
5. How can you use higher derivatives to analyze the motion further?

**Tip**: When solving for velocity or acceleration in kinematics, always check the discriminant of the quadratic equation to determine if real solutions exist.

A particle moves along the x-axis so that at any time t ≥ 0 its position is given by x(t) = t^3 - 3t^2 + 9t - 1. For what values of t is the particle at rest?

Learn how to determine when a particle is at rest using its position function x(t) = t^3 - 3t^2 + 9t - 1. This detailed step-by-step solution involves kinematics, differentiation, and quadratic equations. Suitable for high school students in Grades 10-12.

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