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(a) Which time (or times) represents when the object's speed is at a minimum?
(b) Which time (or times) represents when the object's acceleration is positive?(c) Which time (or times) represents when the object's velocity is negative?

Let's analyze the graph provided to answer each part:

### Observations from the Graph:
- The graph represents the **position vs. time** of an object.
- The slope of the curve indicates the **velocity** of the object (since velocity is the derivative of position with respect to time).
- The curvature of the graph shows the **acceleration** (since acceleration is the derivative of velocity with respect to time).

### (a) Object's Speed at a Minimum:
The speed of the object is the magnitude of the velocity. The speed is at a minimum when the velocity is zero.

1. At point **A**, the slope is positive (increasing).
2. At point **B**, the slope is zero (maximum position, zero velocity).
3. At point **C**, the slope is negative (decreasing).
4. At point **D**, the slope is zero again (minimum position, zero velocity).
5. At point **E**, the slope is positive (increasing).

Therefore, the object's speed is at a minimum at points **B** and **D**.

### (b) Object's Acceleration is Positive:
Acceleration is positive when the slope of the velocity (or concavity of the position vs. time graph) is increasing (curving upwards).

- From **A to B**: The graph is curving upwards (concave up), meaning acceleration is positive.
- From **C to D**: The graph is curving upwards (concave up), meaning acceleration is positive.

Therefore, the acceleration is positive between points **A and B** and between **C and D**.

### (c) Object's Velocity is Negative:
The velocity is negative when the slope of the position vs. time graph is negative.

- Between points **C and D**, the slope is negative (going downwards).

Therefore, the object's velocity is negative between points **C and D**.

### Summary of Answers:
(a) The object's speed is at a minimum at **B** and **D**.  
(b) The object's acceleration is positive between **A and B** and **C and D**.  
(c) The object's velocity is negative between **C and D**.

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How can you determine the acceleration of an object from a position-time graph?
2. What does a negative slope in a position-time graph indicate?
3. How do you calculate the velocity and acceleration from a graph?
4. How is speed different from velocity in physics?
5. What is the significance of concavity in determining acceleration on graphs?

**Tip:** In kinematics, always analyze the slope for velocity and the concavity for acceleration when interpreting position-time graphs.

Explore the analysis of an object's speed, acceleration, and velocity through a position-time graph. Learn when the object's speed is at a minimum, when acceleration is positive, and when velocity is negative with detailed explanations and examples. Suitable for high school students studying kinematics.

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