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at which exact point is instantaneous acceleration  lower than zero

To determine where the instantaneous acceleration is lower than zero on the graph, we need to analyze the slope of the curve. Instantaneous acceleration is given by the derivative of the velocity with respect to time (\(a = \frac{dv}{dt}\)). On a graph of position \(x\) versus time \(t\), acceleration is related to the concavity of the curve:

- When the curve is concave down, acceleration is negative (less than zero).
- When the curve is concave up, acceleration is positive (greater than zero).

### Analysis of the Graph:

Looking at the provided graph, we can see various points labeled from A to H. Let's analyze the concavity at each point:

1. **Point A**: The curve is concave up. Acceleration is positive.
2. **Point B**: The curve is flat (slope is constant). Acceleration is zero.
3. **Point C**: The curve starts to bend downwards (concave down). Acceleration is negative.
4. **Point D**: The curve is again concave down, indicating negative acceleration.
5. **Point E**: The curve is concave up again (the slope starts to increase). Acceleration is positive.
6. **Point F**: The curve is flat (slope is zero). Acceleration is zero.
7. **Point G**: The curve becomes concave down (slope is decreasing). Acceleration is negative.
8. **Point H**: The curve stays flat (zero slope). Acceleration is zero.

### Conclusion:

The points where the instantaneous acceleration is lower than zero (negative) are **Points C, D, and G**, as these are the regions where the curve is concave down.

Would you like more details on any of the points or need further clarification?

### Related Questions:
1. What is the significance of a zero slope in the context of acceleration?
2. How does one determine acceleration from a position-time graph mathematically?
3. What is the difference between instantaneous velocity and instantaneous acceleration?
4. How can you find points of inflection in such graphs?
5. How do changes in acceleration affect the shape of a velocity-time graph?

### Tip:
When analyzing graphs in physics, remember that the slope of a position-time graph gives the velocity, while the slope of a velocity-time graph provides the acceleration.

At which exact point is instantaneous acceleration lower than zero?

Learn how to determine the points on a position-time graph where instantaneous acceleration is lower than zero. This problem covers key calculus concepts, including the derivative of velocity with respect to time and the concavity of curves. Suitable for students in Grades 11-12.

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