Function Analysis: \(f(x) = \frac{x}{\ln x}\)

We are given the function \(f(x) = \frac{x}{\ln x}\), defined for \(x > 1\). We need to evaluate three statements concerning its derivatives and local extrema.

Derivative Calculations for \(f(x)\)

To analyze the function's behavior, we first compute its first and second derivatives using the quotient rule. The quotient rule states that for a function \(f(x) = \frac{g(x)}{h(x)}\), its derivative is \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\).

First Derivative: \(f'(x)\)

For \(f(x) = \frac{x}{\ln x}\), let \(g(x) = x\) and \(h(x) = \ln x\). Their derivatives are \(g'(x) = 1\) and \(h'(x) = \frac{1}{x}\).

Applying the quotient rule:

Second Derivative: \(f''(x)\)

Now, we differentiate \(f'(x)\) using the quotient rule again. Let \(g(x) = \ln x - 1\) and \(h(x) = (\ln x)^2\). Their derivatives are \(g'(x) = \frac{1}{x}\) and \(h'(x) = 2(\ln x) \cdot \frac{1}{x} = \frac{2 \ln x}{x}\).

Applying the quotient rule:

To simplify, we multiply the numerator and the denominator by \(x\):

Expanding the numerator:

Factoring out \(\ln x\) from the numerator and simplifying:

Statement Verification: \(f''(e)\) and Local Minima

Statement 1: \(f''(e) = \frac{1}{e}\)

We substitute \(x=e\) into the expression for \(f''(x)\). We know that \(\ln e = 1\).

Thus, Statement 1 is correct.

Statements 2 and 3: Local Minimum Analysis

To identify potential local extrema, we set the first derivative \(f'(x)\) equal to zero:

This equation is satisfied when the numerator is zero:

The function has a single critical point at \(x = e\). We use the second derivative test to classify this critical point.

We evaluate \(f''(x)\) at \(x=e\). From the verification of Statement 1, we found \(f''(e) = \frac{1}{e}\).

Since \(e \approx 2.718\), \(f''(e) = \frac{1}{e} > 0\). According to the second derivative test, if \(f'(c) = 0\) and \(f''(c) > 0\), the function has a local minimum at \(x=c\). Therefore, \(f(x)\) attains a local minimum value at \(x = e\). This confirms that Statement 2 is correct.

Next, we calculate the value of this local minimum by evaluating \(f(x)\) at \(x=e\):

The local minimum value of \(f(x)\) is \(e\). This confirms that Statement 3 is correct.

Conclusion on Statements

All three statements are verified to be correct. Statement 1 is confirmed by direct calculation of \(f''(e)\). Statements 2 and 3 are confirmed by finding the critical point \(x=e\), using the second derivative test (\(f''(e) > 0\)) to establish it as a local minimum, and determining the minimum value \(f(e) = e\).

Therefore, all three statements are accurate.