The image shows a question asking when a particle is speeding up or slowing down, with answers given in interval notation:

• Speeding up: $0 \geq t \geq 6$
• Slowing down: $t > 6$

However, the intervals presented are incorrect, and they are flagged with warnings.

To solve this problem correctly:

• To determine when a particle is speeding up or slowing down, we need information about the particle's velocity ($v(t)$) and acceleration ($a(t)$). A particle is speeding up when the velocity and acceleration have the same sign (both positive or both negative) and slowing down when they have opposite signs (one positive, one negative).

Without explicit information about the functions, I'll assume the particle's velocity changes over time and reaches a critical point at $t = 6$, and the behavior changes before and after $t = 6$.

1. Speeding up: If $t = 6$ is where velocity reaches zero or changes, and assuming velocity and acceleration have the same sign for $t \leq 6$, the correct interval for speeding up should be something like $(0, 6]$.

2. Slowing down: For $t > 6$, the particle starts to slow down, meaning the interval for slowing down should be $(6, \infty)$.

The correct intervals would look like this:

• Speeding up: $(0, 6]$
• Slowing down: $(6, \infty)$

Let me know if you'd like further explanation, or if you'd like to clarify the velocity and acceleration functions.

Here are five related questions that could deepen your understanding:

1. How do you determine if a particle is speeding up or slowing down based on velocity and acceleration?
2. What does it mean for a particle to have constant acceleration?
3. How can critical points of a velocity function help determine when a particle changes direction?
4. What is the relationship between displacement, velocity, and acceleration in kinematics?
5. How do you interpret graphs of velocity vs. time to analyze a particle's motion?

Tip: Always check if the velocity and acceleration are positive or negative to figure out whether a particle is speeding up or slowing down.