Check whether the Lagrange formula is applicable to following functions :
`f(x) =x^(2) "on" [3,4]`

Answer

To determine whether Lagrange's Mean Value Theorem (LMVT) is applicable to the function \( f(x) = x^2 \) on the interval \([3, 4]\), we will follow these steps:

### Step 1: Check Continuity
The first condition for applying Lagrange's Mean Value Theorem is that the function must be continuous on the closed interval \([a, b]\).

- The function \( f(x) = x^2 \) is a polynomial function, and polynomial functions are continuous everywhere. Therefore, \( f(x) \) is continuous on \([3, 4]\).

### Step 2: Check Differentiability
The second condition is that the function must be differentiable on the open interval \((a, b)\).

- Again, since \( f(x) = x^2 \) is a polynomial function, it is differentiable everywhere. Thus, \( f(x) \) is differentiable on \((3, 4)\).

### Step 3: Apply Lagrange's Mean Value Theorem
Since both conditions are satisfied, we can apply Lagrange's Mean Value Theorem. According to LMVT, there exists at least one point \( c \) in the interval \((3, 4)\) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Where \( a = 3 \) and \( b = 4 \).

### Step 4: Calculate \( f(a) \) and \( f(b) \)
Now we calculate \( f(3) \) and \( f(4) \):

- \( f(3) = 3^2 = 9 \)
- \( f(4) = 4^2 = 16 \)

### Step 5: Compute the Right Side of the Equation
Now we substitute these values into the equation:

\[ f'(c) = \frac{f(4) - f(3)}{4 - 3} = \frac{16 - 9}{1} = \frac{7}{1} = 7 \]

### Step 6: Find \( f'(x) \)
Next, we find the derivative of \( f(x) \):

\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 7: Set Up the Equation
Now we set \( f'(c) = 7 \):

\[ 2c = 7 \]

### Step 8: Solve for \( c \)
Now we solve for \( c \):

\[ c = \frac{7}{2} = 3.5 \]

### Conclusion
Since \( c = 3.5 \) lies within the interval \((3, 4)\), we conclude that Lagrange's Mean Value Theorem is applicable to the function \( f(x) = x^2 \) on the interval \([3, 4]\).

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