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.