Question

Given that \( f(x) = \frac{\log x}{x} \), find the point of local maximum of \( f(x) \).

Answer 1

Step 1: Understand the function
The given function is f(x) = (log x) / x, where log denotes the natural logarithm

Step 2: Calculate the first derivative
Using the quotient rule, the first derivative f'(x) is computed as:
f'(x) = [ (1/x) * x - log x * 1 ] / x² = (1 - log x) / x².

Step 3: Identify critical points
Setting the derivative to zero yields the critical points:
(1 - log x) / x² = 0, which simplifies to 1 - log x = 0.
This implies log x = 1, so x = e (approximately 2.718).

Step 4: Analyze the nature of the critical point
To determine if x = e corresponds to a maximum, we examine the sign changes of f'(x) around x = e.
For values of x just below e, f'(x) is positive (indicating an increasing function).
For values of x just above e, f'(x) is negative (indicating a decreasing function).
Therefore, x = e represents a local maximum.

Step 5: Conclusion
The function f(x) reaches a local maximum at x = e.

Final Answer: x = e