Discuss the applicability of Lagrange's Mean Value Theorem to the following :
`f(x)=x^(2)" on "[2,4]`

Answer

To discuss the applicability of Lagrange's Mean Value Theorem (LMVT) for the function \( f(x) = x^2 \) on the interval \([2, 4]\), we will follow these steps:

### Step 1: Check the conditions of LMVT
The Lagrange's Mean Value Theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

For our function \( f(x) = x^2 \):
- **Continuity**: Since \( f(x) \) is a polynomial function, it is continuous everywhere, including on the interval \([2, 4]\).
- **Differentiability**: Polynomial functions are also differentiable everywhere, including on the interval \((2, 4)\).

### Step 2: Calculate \( f(a) \) and \( f(b) \)
Let \( a = 2 \) and \( b = 4 \).
\[ f(2) = 2^2 = 4 \]
\[ f(4) = 4^2 = 16 \]

### Step 3: Apply the Mean Value Theorem formula
Now we can calculate the right-hand side of the LMVT equation:
\[ \frac{f(b) - f(a)}{b - a} = \frac{f(4) - f(2)}{4 - 2} = \frac{16 - 4}{4 - 2} = \frac{12}{2} = 6 \]

### Step 4: Find \( f'(x) \)
Next, we find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 5: Set \( f'(c) \) equal to the calculated value
We need to find \( c \) such that:
\[ f'(c) = 6 \]
Substituting the expression for \( f'(c) \):
\[ 2c = 6 \]
Solving for \( c \):
\[ c = \frac{6}{2} = 3 \]

### Step 6: Verify that \( c \) is in the interval \((2, 4)\)
Since \( c = 3 \) lies within the interval \((2, 4)\), we conclude that the conditions of the LMVT are satisfied.

### Conclusion
Thus, the Lagrange's Mean Value Theorem is applicable to the function \( f(x) = x^2 \) on the interval \([2, 4]\), and the point \( c \) where the derivative equals the average rate of change is \( c = 3 \).

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