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Let's solve the problem based on the image you uploaded, which concerns Rolle's theorem applied to the function \( f(x) = 2 \sin(x) \) on the interval \([0, 2\pi]\).

### Step 1: Check if Rolle's theorem can be applied.

For Rolle's theorem to apply, three conditions must be met:

1. **Continuity on the closed interval \([a, b]\):**  
   The function \( f(x) = 2 \sin(x) \) is continuous on \([0, 2\pi]\) because the sine function is continuous everywhere.

2. **Differentiability on the open interval \((a, b)\):**  
   The function \( f(x) = 2 \sin(x) \) is differentiable on \((0, 2\pi)\) because the sine function is differentiable everywhere.

3. **\( f(a) = f(b) \):**  
   Here, \( f(0) = 2 \sin(0) = 0 \) and \( f(2\pi) = 2 \sin(2\pi) = 0 \). Thus, \( f(0) = f(2\pi) \), so this condition is also satisfied.

Since all three conditions are satisfied, **Rolle's theorem can be applied**.

### Step 2: Find all values of \( c \) in the open interval \((0, 2\pi)\) such that \( f'(c) = 0 \).

To do this, we first need to compute the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}[2 \sin(x)] = 2 \cos(x).
\]
Now, we set \( f'(x) = 0 \) to find critical points:
\[
2 \cos(x) = 0 \implies \cos(x) = 0.
\]
The cosine function equals 0 at:
\[
x = \frac{\pi}{2}, \frac{3\pi}{2}
\]
in the interval \((0, 2\pi)\).

Thus, the values of \( c \) where \( f'(c) = 0 \) are:
\[
c = \frac{\pi}{2}, \frac{3\pi}{2}.
\]

### Final Answer:
- Rolle's theorem **can be applied**.
- The values of \( c \) are \( \frac{\pi}{2}, \frac{3\pi}{2} \).

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the geometric interpretation of Rolle's theorem?
2. If \( f(x) = x^2 \), can Rolle's theorem be applied on the interval \([-1, 1]\)?
3. What is the derivative of \( f(x) = 3\sin(x) \), and when does it equal zero on \([0, 2\pi]\)?
4. How does Rolle's theorem differ from the Mean Value Theorem?
5. What are the conditions for applying the Mean Value Theorem?

**Tip:** Remember that the function must satisfy all three conditions of Rolle's theorem before you can apply it.

Determine whether Rolle's theorem can be applied to f on the closed interval [a, b]. f(x) = 2sin(x), [0, 2π]. Select all that apply. If Rolle's theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = 0.

This problem explores whether Rolle's theorem can be applied to the function f(x) = 2sin(x) over the interval [0, 2π]. Learn how to check the conditions of Rolle's theorem and calculate critical values c in the open interval where f'(c) = 0. Suitable for students in Grades 11-12.

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