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### 1. Rolle's Theorem Conditions
Rolle's Theorem can be applied if the following conditions are met:
- The function \( f(x) \) is continuous on the closed interval \([a, b]\).
- The function \( f(x) \) is differentiable on the open interval \((a, b)\).
- \( f(a) = f(b) \).

Here, the function is \( f(x) = -x^2 + 9x \) on the interval \([0, 9]\).

#### Step 1: Check if \( f(x) \) is continuous on \([0, 9]\)
\( f(x) = -x^2 + 9x \) is a polynomial function, and polynomial functions are continuous everywhere. Therefore, \( f(x) \) is continuous on the closed interval \([0, 9]\).

#### Step 2: Check if \( f(x) \) is differentiable on \((0, 9)\)
Polynomial functions are also differentiable everywhere. Hence, \( f(x) \) is differentiable on the open interval \((0, 9)\).

#### Step 3: Check if \( f(0) = f(9) \)
- \( f(0) = -(0)^2 + 9(0) = 0 \)
- \( f(9) = -(9)^2 + 9(9) = -81 + 81 = 0 \)

Since \( f(0) = f(9) = 0 \), the third condition is also satisfied.

Therefore, **Rolle's Theorem can be applied**.

### 2. Finding \( c \) such that \( f'(c) = 0 \)
Next, we need to find the value(s) of \( c \) in the open interval \((0, 9)\) where \( f'(c) = 0 \).

- First, find the derivative of \( f(x) \):
  \[
  f'(x) = \frac{d}{dx}(-x^2 + 9x) = -2x + 9
  \]

- Now, set \( f'(x) = 0 \) and solve for \( x \):
  \[
  -2x + 9 = 0
  \]
  \[
  2x = 9
  \]
  \[
  x = \frac{9}{2} = 4.5
  \]

Thus, \( c = 4.5 \) is the value where \( f'(c) = 0 \).

### Answer
- Rolle's Theorem **can** be applied.
- The value of \( c \) is \( 4.5 \).

Would you like more details or have any questions?

Here are some related questions:
1. What are the key conditions for applying Rolle's Theorem?
2. How do you prove that a polynomial is continuous and differentiable?
3. Can you apply Rolle’s Theorem if the function is not differentiable at a point in the interval?
4. What is the geometric interpretation of the value \( c \) in Rolle’s Theorem?
5. How would you approach a function with different behavior (e.g., piecewise) on the interval?

**Tip**: Rolle's Theorem is a special case of the Mean Value Theorem where \( f(a) = f(b) \), ensuring a horizontal tangent line at some point in the interval.

Determine whether Rolle's theorem can be applied to f(x) = -x^2 + 9x on the interval [0, 9], and if so, find the value of c where f'(c) = 0.

Determine if Rolle's theorem can be applied to the function f(x) = -x^2 + 9x on the interval [0, 9], and find the value of c such that f'(c) = 0. Detailed solution and step-by-step explanation provided.

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