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To determine where the function \( f(x) \) is not differentiable based on the graph, we need to identify points where the function has:

1. **Sharp corners or cusps**: The derivative doesn't exist at these points because the slopes from the left and right are different.
2. **Discontinuities**: If the function jumps or has a break, it's not differentiable there.
3. **Vertical tangents**: The slope becomes infinite, so the function is not differentiable.

Looking at the graph you provided:

1. At \( x = -4 \), there is a **corner**.
2. At \( x = 0 \), there is a **discontinuity** (a hole in the graph).
3. At \( x = 2 \), there is a **corner**.

Thus, the function \( f(x) \) is not differentiable at \( x = -4, 0, 2 \).

Would you like further details on differentiability or any specific questions answered? Here are some follow-up questions:

1. What is the difference between a corner and a cusp?
2. Why is the function not differentiable at points with discontinuities?
3. How can vertical tangents affect differentiability?
4. How does the concept of limits relate to differentiability?
5. Can a continuous function always be differentiable?

**Tip**: Differentiability implies continuity, but continuity does not necessarily imply differentiability!

The graph of f is given. State the numbers at which f is not differentiable.

Analyze the differentiability of a function based on its graph. Learn about key points such as corners, discontinuities, and vertical tangents, and understand where a function is not differentiable. Suitable for high school calculus students.

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