Question

The maximum value of the function \( \frac{\log x}{x}, x \neq 0 \) is

• A. \( e^2 \)
• B. \( \frac{1}{e} \)
• C. \( \frac{1}{e^2} \)
• D. \( e \)

Hint:

To find the maximum of a function, differentiate it and set the derivative equal to zero to find the critical points.

Answer 1

The Correct Option is B

Step 1: Maximize the function.
The given function is \( f(x) = \frac{\log x}{x} \). To find the maximum, differentiate with respect to \( x \): \[ f'(x) = \frac{1}{x} \cdot \frac{1}{x} - \frac{\log x}{x^2} \] Set \( f'(x) = 0 \) and solve for \( x \).
Step 2: Conclusion.
The maximum value of \( f(x) \) occurs at \( x = e \), and the maximum value is \( \frac{1}{e} \), corresponding to option (B).