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: Understanding the Concept
Let \(y = (1/x)^x\). Then \(\ln y = -x\ln x\), so maximising \(y\) is the same as maximising \(-x\ln x\).

Step 2: Find the critical point
\[ \frac{d}{dx}(-x\ln x) = -\ln x - 1 = 0 \Rightarrow x = \frac1e \]
The second derivative is \(-\frac1x < 0\) for \(x > 0\), so this is a maximum.

Step 3: Maximum value
\[ \ln y = -\frac1e \ln\frac1e = \frac1e \Rightarrow y_{\max} = e^{1/e} \]
Options (A) and (B) use \(e\) as the exponent, and (D) is less than 1, whereas the function reaches above 1.

Final Answer:
The maximum value is \(e^{1/e}\), option (C). \[ \boxed{e^{1/e}} \]