The minimum value of `(x)/( log x) ` is

Options

A. e
B. `(1)/(e)`
C. `e^(2)`
D. 2e

Answer

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

### Step 1: Differentiate the function
We start by finding the first derivative of \( f(x) \). We will use the quotient rule for differentiation, which states:

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Here, \( u = x \) and \( v = \log x \).

Calculating the derivatives:
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = \frac{1}{x} \)

Now applying the quotient rule:

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

### Step 2: 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:

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

Taking the antilogarithm:

\[ x = e \]

### Step 3: Find the second derivative
Next, we need to determine whether this critical point is a minimum or maximum by finding the second derivative \( f''(x) \).

Using the quotient rule again on \( f'(x) = \frac{\log x - 1}{(\log x)^2} \):

Let \( u = \log x - 1 \) and \( v = (\log x)^2 \).

Calculating the derivatives:
- \( \frac{du}{dx} = \frac{1}{x} \)
- \( \frac{dv}{dx} = 2 \log x \cdot \frac{1}{x} = \frac{2 \log x}{x} \)

Now applying the quotient rule:

\[ f''(x) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Substituting \( u \) and \( v \):

\[ f''(x) = \frac{(\log x)^2 \cdot \frac{1}{x} - (\log x - 1) \cdot \frac{2 \log x}{x}}{(\log x)^4} \]

### Step 4: Evaluate the second derivative at \( x = e \)
Now we evaluate \( f''(x) \) at \( x = e \):

Since \( \log e = 1 \):

\[ f''(e) = \frac{(1)^2 \cdot \frac{1}{e} - (1 - 1) \cdot \frac{2 \cdot 1}{e}}{(1)^4} = \frac{\frac{1}{e}}{1} = \frac{1}{e} \]

Since \( f''(e) > 0 \), this indicates that \( x = e \) is a point of local minimum.

### Step 5: Find the minimum value
Finally, we find the minimum value of \( f(x) \) at \( x = e \):

\[ f(e) = \frac{e}{\log e} = \frac{e}{1} = e \]

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

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