From the provided graph, I will analyze the differentiability of the function at key points. Differentiability fails where the graph shows any sharp turns (cusps or corners), discontinuities, or vertical tangents.

Points where the function fails to be differentiable:

1. x = -10:

   • The graph has a sharp cusp here, meaning that the slope of the tangent line changes abruptly. The function is continuous but not differentiable because the derivative (slope) from the left is different from the right.

2. x = -4:

   • There is a jump discontinuity at $x = -4$. The graph breaks and moves directly to a different value. At points of discontinuity, the function is not continuous, so it cannot be differentiable.

3. x = 0:

   • Another sharp corner (cusp) appears at $x = 0$, similar to $x = -10$, where the slope from either side of the point is different, causing the function to fail to be differentiable.

4. x = 8:

   • There's a cusp at $x = 8$, where the graph turns sharply. The tangent slopes on either side are not equal, indicating the function is not differentiable here.

Key Point: $x = 10$

• The graph at $x = 10$ does not show any issues with continuity or sharp turns, so there is no problem with differentiability at this point.

Conclusion

The function fails to be differentiable at:

• $x = -10$
• $x = -4$
• $x = 0$
• $x = 8$

Would you like more details or have any questions?

Related Questions:

1. What are the conditions for a function to be differentiable at a point?
2. How does continuity relate to differentiability?
3. Can you have a function that is continuous but not differentiable at a point?
4. What is the significance of sharp turns or cusps in graphs?
5. How do vertical tangents affect the differentiability of a function?

Tip:

For a function to be differentiable at a point, it must be both continuous and smooth at that point, meaning no sharp corners, cusps, or discontinuities.