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?

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.