Question

Find the intervals in which the function \( f(x) = \frac{\log x}{x} \) is strictly increasing or strictly decreasing.

Hint:

To find increasing or decreasing intervals, analyze the sign of \( f'(x) \) over different domains.

Answer 1

1. Derivative Calculation: The derivative of \( f(x) = \frac{\log x}{x} \) is computed as \( f'(x) = \frac{1 \cdot x - \log x \cdot 1}{x^2} = \frac{x - \log x}{x^2}. \)
2. Identifying Critical Points: Critical points are found by setting \( f'(x) = 0 \), which leads to \( x - \log x = 0 \), or \( x = \log x \). This equation is satisfied when \( x = e \). 
3. Analysis of \( f'(x) \) Sign: - For \( x \in (0, e) \), the expression \( x - \log x \) is positive, meaning \( f'(x)>0 \). Consequently, \( f(x) \) is increasing in this interval. - For \( x \in (e, \infty) \), the expression \( x - \log x \) is negative, resulting in \( f'(x)<0 \). Therefore, \( f(x) \) is decreasing in this interval. 
Monotonicity Intervals: \( f(x) \) exhibits strict increase on \( (0, e) \) and strict decrease on \( (e, \infty) \).