Verify Lagrange's Mean Value Theorem for the functions :
`f(x)=sqrt(x^(2)-4)` in the interval [2, 4].

Answer

To verify Lagrange's Mean Value Theorem (LMVT) for the function \( f(x) = \sqrt{x^2 - 4} \) on the interval \([2, 4]\), we will follow these steps:

### Step 1: Check if the function is continuous on the interval [2, 4]
The function \( f(x) = \sqrt{x^2 - 4} \) is defined for \( x^2 - 4 \geq 0 \), which simplifies to \( x \geq 2 \) or \( x \leq -2 \). Therefore, the function is defined and continuous for \( x \in [2, 4] \).

### Step 2: Check if the function is differentiable on the interval (2, 4)
To check differentiability, we need to find the derivative of \( f(x) \). The derivative is given by:
\[ f'(x) = \frac{d}{dx}(\sqrt{x^2 - 4}) = \frac{1}{2\sqrt{x^2 - 4}} \cdot (2x) = \frac{x}{\sqrt{x^2 - 4}} \]
This derivative exists for all \( x \) in the interval \( (2, 4) \) since \( \sqrt{x^2 - 4} \) is positive for \( x > 2 \). Hence, \( f(x) \) is differentiable on \( (2, 4) \).

### Step 3: Calculate \( f(a) \) and \( f(b) \)
Let \( a = 2 \) and \( b = 4 \).
\[ f(2) = \sqrt{2^2 - 4} = \sqrt{0} = 0 \]
\[ f(4) = \sqrt{4^2 - 4} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \]

### Step 4: Apply the Mean Value Theorem
According to LMVT, there exists at least one \( c \in (2, 4) \) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
Substituting the values we calculated:
\[ f'(c) = \frac{2\sqrt{3} - 0}{4 - 2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Step 5: Set the derivative equal to the average rate of change
We have:
\[ f'(c) = \frac{c}{\sqrt{c^2 - 4}} = \sqrt{3} \]
Squaring both sides:
\[ \frac{c^2}{c^2 - 4} = 3 \]
Cross-multiplying gives:
\[ c^2 = 3(c^2 - 4) \implies c^2 = 3c^2 - 12 \implies 2c^2 = 12 \implies c^2 = 6 \implies c = \sqrt{6} \]

### Step 6: Verify that \( c \) is in the interval (2, 4)
Since \( \sqrt{6} \) is approximately 2.45, which lies within the interval (2, 4), we have found a valid \( c \).

### Conclusion
Thus, we have verified Lagrange's Mean Value Theorem for the function \( f(x) = \sqrt{x^2 - 4} \) on the interval \([2, 4]\).

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