Verify Langrange's mean value theorem for the function :
f(x)=x(1-log x) and find the value of 'c' in the interval [1,2].

Answer

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

### Step 1: Check continuity and differentiability
The function \( f(x) = x(1 - \log x) \) is a product of continuous functions (polynomial and logarithmic functions) in the interval \([1, 2]\). Since logarithmic functions are continuous for \( x > 0 \), \( f(x) \) is continuous on \([1, 2]\) and differentiable on \((1, 2)\).

**Hint:** Ensure that the function is continuous and differentiable in the given interval.

### Step 2: Calculate \( f(1) \) and \( f(2) \)
Now we calculate the values of the function at the endpoints of the interval.
- \( f(1) = 1(1 - \log 1) = 1(1 - 0) = 1 \)
- \( f(2) = 2(1 - \log 2) = 2(1 - \log 2) \)

**Hint:** Substitute the values of the endpoints into the function to find \( f(1) \) and \( f(2) \).

### Step 3: Find the average rate of change
Using the values calculated:
\[ \text{Average rate of change} = \frac{f(2) - f(1)}{2 - 1} = f(2) - 1 = 2(1 - \log 2) - 1 \]

**Hint:** Calculate the average rate of change using the formula for the slope of the secant line between the two points.

### Step 4: Differentiate \( f(x) \)
Next, we differentiate \( f(x) \):
\[ f(x) = x(1 - \log x) \implies f'(x) = 1 - \log x - 1 = -\log x \]

**Hint:** Use the product rule to differentiate the function.

### Step 5: Apply Lagrange's Mean Value Theorem
According to LMVT, there exists at least one \( c \in (1, 2) \) such that:
\[ f'(c) = \frac{f(2) - f(1)}{2 - 1} \]
This means:
\[ -\log c = 2(1 - \log 2) - 1 \]

**Hint:** Set the derivative equal to the average rate of change to find \( c \).

### Step 6: Solve for \( c \)
From the equation:
\[ -\log c = 2 - 2\log 2 - 1 \implies -\log c = 1 - 2\log 2 \]
Thus,
\[ \log c = 2\log 2 - 1 \implies c = e^{2\log 2 - 1} = \frac{4}{e} \]

**Hint:** Use properties of logarithms and exponentials to isolate \( c \).

### Step 7: Verify \( c \) is in the interval \([1, 2]\)
We know \( e \approx 2.718 \), hence \( \frac{4}{e} \approx \frac{4}{2.718} \approx 1.47 \), which lies in the interval \( [1, 2] \).

**Hint:** Check that the calculated \( c \) value falls within the specified interval.

### Conclusion
We have verified Lagrange's Mean Value Theorem for the function \( f(x) = x(1 - \log x) \) in the interval \([1, 2]\) and found that \( c = \frac{4}{e} \).