Check whether the Lagrange formula is applicable to following functions :
`f(x)= " ln x on " [1,3]`

Answer

To determine whether Lagrange's Mean Value Theorem (LMVT) is applicable to the function \( f(x) = \ln x \) on the interval \([1, 3]\), we need to follow these steps:

### Step 1: Check Continuity
The first condition for LMVT is that the function must be continuous on the closed interval \([a, b]\).

- **Function**: \( f(x) = \ln x \)
- **Interval**: \([1, 3]\)

The natural logarithm function \( \ln x \) is continuous for all \( x > 0 \). Since the interval \([1, 3]\) lies within this domain, \( f(x) \) is continuous on \([1, 3]\).

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

- The function \( f(x) = \ln x \) is differentiable for all \( x > 0 \). Thus, it is differentiable on the open interval \((1, 3)\).

### Step 3: Apply Lagrange's Mean Value Theorem
Since both conditions are satisfied, we can apply LMVT. According to LMVT, there exists at least one \( c \) in the interval \((1, 3)\) such that:

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

Where \( a = 1 \) and \( b = 3 \).

### Step 4: Calculate \( f(1) \) and \( f(3) \)
- \( f(1) = \ln(1) = 0 \)
- \( f(3) = \ln(3) \)

### Step 5: Compute the Right-Hand Side
Now we compute:

\[ \frac{f(3) - f(1)}{3 - 1} = \frac{\ln(3) - 0}{3 - 1} = \frac{\ln(3)}{2} \]

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

\[ f'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} \]

### Step 7: Set Up the Equation
According to LMVT, we need to find \( c \) such that:

\[ f'(c) = \frac{\ln(3)}{2} \]

Substituting \( f'(c) \):

\[ \frac{1}{c} = \frac{\ln(3)}{2} \]

### Step 8: Solve for \( c \)
To find \( c \), we rearrange the equation:

\[ c = \frac{2}{\ln(3)} \]

### Conclusion
Thus, since we have verified that \( f(x) \) is continuous on \([1, 3]\) and differentiable on \((1, 3)\), and we found a \( c \) that satisfies the mean value theorem, we conclude that Lagrange's Mean Value Theorem is applicable to the function \( f(x) = \ln x \) on the interval \([1, 3]\).