The x y-coordinate plane is given. A curve and a point are on the graph.
The curve enters the window in the second quadrant, goes up and right, changes direction in the second quadrant, goes down and right, passes through an open point in the second quadrant above x = −1, crosses the x-axis at approximately x = −0.5, crosses the negative y-axis, changes direction in the fourth quadrant, goes up and right, crosses the x-axis at approximately x = 1.6, sharply changes direction in the first quadrant above x = 2, goes down and right, crosses the x-axis at approximately x = 2.4, and exits the window in the fourth quadrant.
The closed point is located in the second quadrant, directly above the open point above x = −1.

The graph of f is given. State the numbers at which f is not differentiable. (Enter your answers as a comma-separated list.)

x=

To determine the points at which \( f(x) \) is not differentiable, we need to focus on a few key aspects of the graph, based on the description provided:

1. **Sharp Turns or Corners**: Differentiability fails at points where the curve has a sharp turn. From the description, the curve "sharply changes direction" at two points: in the second quadrant (just above \( x = -1 \)) and in the first quadrant above \( x = 2 \). These are likely points where differentiability fails.

2. **Open Point**: There is an open point in the second quadrant above \( x = -1 \). This suggests a discontinuity, and functions are not differentiable at points of discontinuity. This is another possible point where \( f(x) \) is not differentiable.

3. **Endpoints**: If the curve exits or enters the graph window, we generally don't consider those as points where the function is defined, so they don’t directly affect differentiability unless we are given limits.

From this analysis, we can infer that the function \( f(x) \) is not differentiable at:

- \( x = -1 \) (due to the open point and sharp turn)
- \( x = 2 \) (sharp change in direction)

Thus, the points at which \( f(x) \) is not differentiable are:
\[ x = -1, 2 \]

Let me know if you'd like more details or explanations! 

### Further questions to explore:
1. What is the graphical condition for differentiability?
2. How do we identify a point of discontinuity from a graph?
3. Why does a sharp turn imply non-differentiability?
4. Can a function be continuous but not differentiable? Provide examples.
5. What happens to the slope of a tangent at points of sharp turns?

**Tip**: Always remember that differentiability implies continuity, but continuity does not necessarily imply differentiability.

The x y-coordinate plane is given. A curve and a point are on the graph. The curve enters the window in the second quadrant, goes up and right, changes direction in the second quadrant, goes down and right, passes through an open point in the second quadrant above x = −1, crosses the x-axis at approximately x = −0.5, crosses the negative y-axis, changes direction in the fourth quadrant, goes up and right, crosses the x-axis at approximately x = 1.6, sharply changes direction in the first quadrant above x = 2, goes down and right, crosses the x-axis at approximately x = 2.4, and exits the window in the fourth quadrant. The closed point is located in the second quadrant, directly above the open point above x = −1. The graph of f is given. State the numbers at which f is not differentiable.

Learn how to identify points of non-differentiability on a curve in the coordinate plane. This problem covers key calculus concepts, such as differentiability, sharp turns, and discontinuities, with a detailed solution involving graphical interpretation. Suitable for students in Grades 10-12.

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