The minimum value of `x/(logx)` is:

Options

A. e
B. `1/e`
C. 1
D. None of these

Answer

To find the minimum value of the function \( F(x) = \frac{x}{\log x} \), we will follow these steps:

### Step 1: Define the function
Let \( F(x) = \frac{x}{\log x} \).

### Step 2: Differentiate the function
To find the critical points, we need to differentiate \( F(x) \). We will use the quotient rule for differentiation, which states that if \( F(x) = \frac{U}{V} \), then:

\[ F'(x) = \frac{U'V - UV'}{V^2} \]

Here, \( U = x \) and \( V = \log x \).

Calculating the derivatives:
- \( U' = 1 \)
- \( V' = \frac{1}{x} \)

Now applying the quotient rule:

\[ F'(x) = \frac{(1)(\log x) - (x)(\frac{1}{x})}{(\log x)^2} = \frac{\log x - 1}{(\log x)^2} \]

### Step 3: Set the derivative to zero
To find the critical points, we set \( F'(x) = 0 \):

\[ \frac{\log x - 1}{(\log x)^2} = 0 \]

This implies that:

\[ \log x - 1 = 0 \implies \log x = 1 \]

### Step 4: Solve for \( x \)
From \( \log x = 1 \), we can exponentiate both sides to find \( x \):

\[ x = e \]

### Step 5: Verify if it is a minimum
To confirm that this critical point is a minimum, we can check the second derivative or analyze the sign of the first derivative around \( x = e \). However, since we are looking for the minimum value, we can directly evaluate \( F(x) \) at \( x = e \).

### Step 6: Calculate the minimum value
Now we substitute \( x = e \) into the original function:

\[ F(e) = \frac{e}{\log e} \]

Since \( \log e = 1 \):

\[ F(e) = \frac{e}{1} = e \]

### Conclusion
Thus, the minimum value of \( \frac{x}{\log x} \) is \( e \).