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: Find the first derivative
To find local maxima, we first differentiate f(x) using the quotient rule:
f'(x) = [ (1/x) * x - log x * 1 ] / x² = (1 - log x) / x².

Step 3: Find critical points
Set the derivative equal to zero to find critical points:
(1 - log x) / x² = 0 implies 1 - log x = 0
=> log x = 1
=> x = e (where e ≈ 2.718)

Step 4: Determine the nature of critical point
To check if x = e is a maximum, examine the second derivative or test values around x = e.
For x slightly less than e, f'(x) > 0 (function increasing).
For x slightly greater than e, f'(x) < 0 (function decreasing).
Hence, x = e is a point of local maximum.

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

Final Answer: x = e