To determine where the function is not differentiable, we need to look for certain characteristics in the graph. A function is not differentiable at points where:

1. There is a sharp corner or cusp.
2. The function is discontinuous (a jump or a hole in the graph).
3. The slope is vertical (the tangent line has an infinite slope).

From the graph:

• At $x = -4$: There is a sharp corner, which means the function is not differentiable at this point.
• At $x = -2$: There is a jump discontinuity (a gap in the function), so the function is not differentiable here.
• At $x = 0$: There is a hole in the graph, indicating a discontinuity, so the function is not differentiable at this point.
• At $x = 2$: There is a sharp corner, so the function is not differentiable at this point.

Conclusion:

The function is not differentiable at $x = -4, -2, 0, 2$.

Would you like further explanation or details on why differentiability fails at these points?

Here are 5 related questions to expand on this topic:

1. What is the formal definition of differentiability at a point?
2. How do continuous functions differ from differentiable functions?
3. Can a function be continuous but not differentiable at certain points?
4. What are the implications of having vertical tangents on differentiability?
5. How do limits play a role in determining differentiability?

Tip: If a function has a sharp corner or cusp, it's an indication that the left-hand and right-hand slopes do not match, hence it's not differentiable at that point.