Rolle's Theorem for the following function `f(x)= x sqrt(4-x^(2))`, in [0, 2] is vertified, then find c

Answer

To verify Rolle's Theorem for the function \( f(x) = x \sqrt{4 - x^2} \) on the interval \([0, 2]\) and find the value of \( c \), we will follow these steps:

### Step 1: Check the conditions of Rolle's Theorem
Rolle's Theorem states that if a function is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

1. **Continuity**: The function \( f(x) = x \sqrt{4 - x^2} \) is a product of continuous functions (since \( \sqrt{4 - x^2} \) is continuous for \( x \) in \([-2, 2]\)).
2. **Differentiability**: The function is differentiable in the open interval \((0, 2)\).
3. **Equal values at endpoints**: Calculate \( f(0) \) and \( f(2) \):
- \( f(0) = 0 \cdot \sqrt{4 - 0^2} = 0 \)
- \( f(2) = 2 \cdot \sqrt{4 - 2^2} = 2 \cdot \sqrt{0} = 0 \)

Since \( f(0) = f(2) = 0 \), all conditions of Rolle's Theorem are satisfied.

### Step 2: Find the derivative \( f'(x) \)
Using the product rule, where \( u = x \) and \( v = \sqrt{4 - x^2} \):
\[ f'(x) = u'v + uv' \]
Where:
- \( u' = 1 \)
- \( v = (4 - x^2)^{1/2} \)
- \( v' = \frac{1}{2}(4 - x^2)^{-1/2} \cdot (-2x) = -\frac{x}{\sqrt{4 - x^2}} \)

Thus,
\[ f'(x) = 1 \cdot \sqrt{4 - x^2} + x \cdot \left(-\frac{x}{\sqrt{4 - x^2}}\right) \]
\[ f'(x) = \sqrt{4 - x^2} - \frac{x^2}{\sqrt{4 - x^2}} \]
Combining the terms:
\[ f'(x) = \frac{(4 - x^2) - x^2}{\sqrt{4 - x^2}} = \frac{4 - 2x^2}{\sqrt{4 - x^2}} \]

### Step 3: Set the derivative equal to zero and solve for \( c \)
Set \( f'(c) = 0 \):
\[ \frac{4 - 2c^2}{\sqrt{4 - c^2}} = 0 \]
This implies:
\[ 4 - 2c^2 = 0 \]
Solving for \( c^2 \):
\[ 2c^2 = 4 \implies c^2 = 2 \implies c = \pm \sqrt{2} \]

### Step 4: Determine the valid value of \( c \)
Since \( c \) must lie in the interval \((0, 2)\), we have:
\[ c = \sqrt{2} \]

### Final Answer
Thus, the value of \( c \) that satisfies Rolle's Theorem for the function \( f(x) = x \sqrt{4 - x^2} \) in the interval \([0, 2]\) is:
\[ \boxed{\sqrt{2}} \]