Let \( f(x)=\frac{\ln x}{x} \), determine whether a relative maximum or minimum there exists.

To determine whether the function \( f(x) = \frac{\ln x}{x} \) has any relative maxima or minima, we'll perform a critical point analysis. ### 1. **Find the First Derivative** First, compute the derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx} \left( \frac{\ln x}{x} \right) \] Using the quotient rule: \[ f'(x) = \frac{\frac{1}{x} \cdot x - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \] ### 2. **Find Critical Points** Set \( f'(x) = 0 \) to find critical points: \[ \frac{1 - \ln x}{x^2} = 0 \implies 1 - \ln x = 0 \implies \ln x = 1 \implies x = e \] So, \( x = e \) is the only critical point. ### 3. **Determine the Nature of the Critical Point** To ascertain whether this critical point is a maximum or minimum, examine the sign of \( f'(x) \) around \( x = e \): - **For \( x < e \):** - \( \ln x < 1 \) (since \( \ln e = 1 \)) - \( 1 - \ln x > 0 \) - \( f'(x) > 0 \) - **For \( x > e \):** - \( \ln x > 1 \) - \( 1 - \ln x < 0 \) - \( f'(x) < 0 \) Since \( f'(x) \) changes from positive to negative as \( x \) passes through \( e \), the function \( f(x) \) attains a **relative maximum** at \( x = e \). ### 4. **Value of the Relative Maximum** Compute \( f(e) \): \[ f(e) = \frac{\ln e}{e} = \frac{1}{e} \] ### 5. **Conclusion** - **Relative Maximum:** At \( x = e \), with \( f(e) = \frac{1}{e} \). - **No Relative Minimum:** There are no other critical points where a relative minimum could occur. #### **Graphical Insight** The function increases on the interval \( (0, e) \) and decreases on \( (e, \infty) \), confirming the presence of a single relative maximum at \( x = e \). **Final Answer:** All relative extrema of f  occur at x = e, where f has a relative maximum. The function has no relative minimum.