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Let's go through the problem step-by-step to determine if Rolle's theorem can be applied and, if it can, find the value(s) of \( c \) where \( f'(c) = 0 \).

### Step 1: Understanding Rolle's Theorem Requirements
Rolle's theorem applies to a function \( f(x) \) on a closed interval \([a, b]\) if:
1. **Continuity on the closed interval \([a, b]\)**.
2. **Differentiability on the open interval \((a, b)\)**.
3. **Equal values at the endpoints**: \( f(a) = f(b) \).

Given:
- \( f(x) = -x^2 + 2x \)
- Interval \([0, 2]\)

We will verify if \( f(x) \) satisfies these conditions.

### Step 2: Verify Continuity
The function \( f(x) = -x^2 + 2x \) is a polynomial, and polynomials are continuous everywhere. Therefore, \( f(x) \) is continuous on \([0, 2]\).

### Step 3: Verify Differentiability
Since \( f(x) \) is a polynomial, it is also differentiable everywhere, including on the open interval \((0, 2)\).

### Step 4: Check if \( f(0) = f(2) \)
Now, let's calculate \( f(0) \) and \( f(2) \):
- \( f(0) = -(0)^2 + 2 \cdot 0 = 0 \)
- \( f(2) = -(2)^2 + 2 \cdot 2 = -4 + 4 = 0 \)

Since \( f(0) = f(2) = 0 \), the third condition is also satisfied.

### Conclusion
All three conditions of Rolle's theorem are met, so **Rolle's theorem can be applied**.

### Step 5: Find \( c \) such that \( f'(c) = 0 \)
1. First, compute \( f'(x) \):
   \[
   f(x) = -x^2 + 2x
   \]
   \[
   f'(x) = -2x + 2
   \]
2. Set \( f'(x) = 0 \) to find the value of \( c \):
   \[
   -2x + 2 = 0
   \]
   \[
   2x = 2
   \]
   \[
   x = 1
   \]

So, \( c = 1 \) is the value in the open interval \((0, 2)\) where \( f'(c) = 0 \).

### Final Answer
- **Rolle's theorem can be applied** (select the first option).
- The value of \( c \) is \( 1 \).

Would you like further details or explanations on any specific part of this problem?

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Here are five related questions to deepen your understanding:

1. What are the consequences if \( f(x) \) were not continuous on \([a, b]\)?
2. How would you solve this if the function were not differentiable on \((a, b)\)?
3. Can Rolle's theorem be applied to functions that have discontinuities or cusps?
4. How does Rolle's theorem relate to the Mean Value Theorem?
5. What if \( f(a) \neq f(b) \); how does this affect the theorem’s application?

**Tip:** Always ensure that a function meets all criteria before applying theorems like Rolle's.

Determine whether Rolle's theorem can be applied to f on the closed interval [a, b]. Select all that apply. Given: f(x) = -x^2 + 2x on the interval [0, 2].

Explore the application of Rolle's theorem to the function f(x) = -x^2 + 2x over the interval [0, 2]. Verify continuity, differentiability, and equal endpoint values, and find the value of c where f'(c) = 0.

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