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State with reasons the points at which the function is not differentiable

To determine the points at which the function in the given graph is not differentiable, we need to examine the behavior of the graph in terms of sharp corners, discontinuities, and vertical tangents. A function is not differentiable at any point where one of these conditions occurs:

1. **Discontinuities**: If the function has a break, jump, or hole, it is not differentiable at those points.
2. **Sharp Corners/Cusps**: If the function has a sharp turn (cusp), it is not differentiable at those points, as the derivative does not exist due to the abrupt change in direction.
3. **Vertical Tangents**: If the slope becomes vertical (undefined), the derivative does not exist.

### Analysis of the Graph:

- **At \(x = -1\)**: The graph has a sharp corner where the slope changes abruptly, so the function is not differentiable at this point.
  
- **At \(x = 0\)**: The graph has a hole, indicating a discontinuity. Since the function is discontinuous, it is not differentiable at \(x = 0\).

- **At \(x = 2\)**: There is another hole in the graph, meaning a discontinuity at this point. Thus, the function is not differentiable at \(x = 2\).

- **At \(x = 3\)**: The graph has a sharp corner at this point as well, indicating a non-differentiable point due to the abrupt change in slope.

### Points of Non-Differentiability:
1. \( x = -1 \) (sharp corner)
2. \( x = 0 \) (discontinuity)
3. \( x = 2 \) (discontinuity)
4. \( x = 3 \) (sharp corner)

Would you like further explanation on any of these points, or do you have any questions?

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Here are 5 relative questions to expand your understanding:

1. Why is differentiability closely linked to continuity in a function?
2. Can a function be continuous but not differentiable? If so, why?
3. What is the geometric interpretation of a derivative at a point?
4. How can limits help in determining differentiability at a point?
5. What is the relationship between differentiability and smoothness of a graph?

**Tip**: Always check for both continuity and smoothness when identifying differentiable points on a graph.

Determine the points at which a function is not differentiable by analyzing sharp corners and discontinuities. This problem covers key concepts in calculus, including differentiability, continuity, and graphical interpretation, suitable for students in Grades 11-12.

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