Question

The maximum value of $f(x) = \frac{\log x{x}$ occurs at $x = $:

• A. 1
• B. $e$
• C. $1/e$
• D. 2

Hint:

$\log x$ is natural log ($\ln x$) in calculus unless specified otherwise.

Answer 1

The Correct Option is B

Step 1: Concept
Find $f'(x)$ and set it to zero for critical points.
Step 2: Analysis
$f'(x) = \frac{x(1/x) - \log x(1)}{x^{2}} = \frac{1 - \log x}{x^{2}}$.
$1 - \log x = 0 \Rightarrow \log x = 1 \Rightarrow x = e$.
Step 3: Conclusion
At $x=e$, $f'(x)$ changes from positive to negative, confirming a maximum.
Final Answer: (B)