Verify Lagrange's Mean Value Theorem for the functions :
`f(x)=log_(e)x` in the interval [1, 2]

Answer

To verify Lagrange's Mean Value Theorem (LMVT) for the function \( f(x) = \log_e x \) in the interval \([1, 2]\), we will follow these steps:

### Step 1: Check the conditions of LMVT
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 \( c \) in \((a, b)\) such that:

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

For our function \( f(x) = \log_e x \):
- The function is continuous on \([1, 2]\) because the logarithm function is continuous for all \( x > 0 \).
- The function is differentiable on \((1, 2)\) because the logarithm function is differentiable for all \( x > 0 \).

### Step 2: Calculate \( f(a) \) and \( f(b) \)
Let \( a = 1 \) and \( b = 2 \).

\[ f(1) = \log_e(1) = 0 \]
\[ f(2) = \log_e(2) \]

### Step 3: Calculate the average rate of change
Now, we calculate the average rate of change of the function over the interval \([1, 2]\):

\[ \frac{f(b) - f(a)}{b - a} = \frac{f(2) - f(1)}{2 - 1} = \frac{\log_e(2) - 0}{1} = \log_e(2) \]

### Step 4: Find \( f'(x) \)
Next, we need to find the derivative of the function:

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

### Step 5: Set \( f'(c) \) equal to the average rate of change
According to LMVT, there exists a \( c \in (1, 2) \) such that:

\[ f'(c) = \log_e(2) \]

Substituting \( f'(c) \):

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

### Step 6: Solve for \( c \)
Now, we solve for \( c \):

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

### Step 7: Verify that \( c \) is in the interval \((1, 2)\)
To ensure that \( c \) lies within the interval \((1, 2)\), we need to check:

1. Since \( \log_e(2) \) is a positive number (approximately 0.693), \( c = \frac{1}{\log_e(2)} \) will be greater than 1.
2. We also need to check if \( c < 2 \):

\[ \frac{1}{\log_e(2)} < 2 \implies 1 < 2 \log_e(2) \implies \frac{1}{2} < \log_e(2) \]

This inequality holds true since \( \log_e(2) \approx 0.693 \).

### Conclusion
Thus, we have verified that there exists at least one \( c \in (1, 2) \) such that \( f'(c) = \log_e(2) \), satisfying the conditions of Lagrange's Mean Value Theorem.

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