Question

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

Hint:

To determine monotonicity, find \( f'(x) \), solve \( f'(x) = 0 \), and test the sign of \( f'(x) \) in intervals.

Answer 1

Step 1: Find the derivative of \( f(x) \)
The given function is \( f(x) = \frac{\log x}{x} \). Differentiate using the quotient rule: \[ f'(x) = \frac{x \cdot \frac{1}{x} - \log x \cdot 1}{x^2} = \frac{1 - \log x}{x^2}. \] 
Step 2: Find critical points
For \( f'(x) = 0 \): \[ 1 - \log x = 0 \implies \log x = 1 \implies x = e. \] 
Step 3: Determine intervals of increase and decrease
For \( x \in (0, e) \): \[ 1 - \log x>0 \implies f'(x)>0 \quad {(strictly increasing)}. \] For \( x \in (e, \infty) \): \[ 1 - \log x<0 \implies f'(x)<0 \quad {(strictly decreasing)}. \] 
Step 4: Conclude the result
The function is strictly increasing on \( (0, e) \) and strictly decreasing on \( (e, \infty) \).