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The question asks to determine the values of \( c \) that are guaranteed by Rolle's Theorem for the given function \( f(x) \) over the interval \([-4, 6]\).

### Rolle's Theorem Conditions:
For Rolle's Theorem to apply, the function \( f(x) \) must meet three conditions:
1. **Continuity**: \( f(x) \) is continuous on the closed interval \([-4, 6]\).
2. **Differentiability**: \( f(x) \) is differentiable on the open interval \((-4, 6)\).
3. **Equal End Values**: \( f(-4) = f(6) \).

Once these conditions are met, Rolle's Theorem guarantees that there exists at least one \( c \) in the interval \((-4, 6)\) such that \( f'(c) = 0 \), i.e., a point where the tangent line is horizontal.

### Step-by-Step:
1. **Check for continuity and differentiability**: The graph appears to be smooth (no breaks or sharp turns), indicating that the function is continuous and differentiable over the given interval.
2. **Equal End Values**: From the graph, we need to check whether \( f(-4) = f(6) \). Both \( f(-4) \) and \( f(6) \) are equal to 0, so the third condition is met.

Thus, Rolle's Theorem applies, and we need to find the values of \( c \) where \( f'(c) = 0 \) (these are the points where the graph has horizontal tangents, or the critical points).

### Critical Points (where slope is zero):
By visually inspecting the graph, the critical points where the slope of the graph is zero (i.e., horizontal tangent lines) occur approximately at:
- \( c = -2 \)
- \( c = 2 \)

These are the points where the function reaches local maxima or minima.

### Final Answer:
The values of \( c \) that satisfy Rolle's Theorem are \( c = -2, 2 \).

Do you have any questions or would you like further details?

Here are 5 related questions for practice:
1. How is Rolle's Theorem different from the Mean Value Theorem?
2. What are the necessary conditions for the Mean Value Theorem to apply?
3. Can a function be continuous but not differentiable? Provide an example.
4. What is the geometric interpretation of Rolle's Theorem?
5. How do we find the derivative of a function from its graph?

**Tip**: Always ensure that the function satisfies all three conditions of Rolle's Theorem before applying it to find the critical points.

Given the graph of f(x), determine all values of c guaranteed by Rolle's Theorem using the interval [-4,6].

This problem involves using Rolle's Theorem to determine the values of c from the given graph of f(x) over the interval [-4, 6]. Key concepts include continuity, differentiability, and identifying critical points. Suitable for students in Grades 11-12 or early university.

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