Question

The maximum value of \((\frac{1}{x})^x\), \(x > 0\) is

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

Hint:

Take the log of the function and set the derivative to zero.

Answer 1

The Correct Option is C

Step 1: Take log
$\ln y = -x \ln x$ for $x > 0$.

Step 2: Optimise
Setting the derivative $-\ln x - 1$ to zero gives $x = 1/e$, and the sign changes from positive to negative, so it is a maximum.

Step 3: Value
$\ln y = \frac{1}{e}$, hence $y = e^{1/e}$. Option (C).

Final Answer:
Option (C). \[ \boxed{e^{1/e}} \]