Question

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

• A. \( e^{1/e} \)
• B. \( e^e \)
• C. \( 1 \)
• D. Infinity
• E. \( 0 \)

Hint:

For functions like \( x^x \) or \( (1/x)^x \), always use logarithmic differentiation.

Answer 1

The Correct Option is A

Concept: To find the maximum of a function involving a variable in both base and exponent, we take logarithm and then differentiate.

Step 1: Take logarithm. Given: \[ y = \left(\frac{1}{x}\right)^x \] Take natural logarithm: \[ \ln y = x \ln\left(\frac{1}{x}\right) \] \[ = x (-\ln x) = -x \ln x \]

Step 2: Differentiate. Differentiate both sides w.r.t. \(x\): \[ \frac{1}{y}\frac{dy}{dx} = -\left(\ln x + 1\right) \] (Using product rule: \( \frac{d}{dx}(x\ln x) = \ln x + 1 \)) \[ \Rightarrow \frac{dy}{dx} = y(-\ln x - 1) \]

Step 3: Find critical point. For maxima/minima: \[ \frac{dy}{dx} = 0 \] Since \(y \neq 0\), we get: \[ -\ln x - 1 = 0 \Rightarrow \ln x = -1 \Rightarrow x = e^{-1} = \frac{1}{e} \]

Step 4: Check maximum. Second derivative (or sign change): \[ \frac{dy}{dx} \text{ changes from positive to negative at } x=\frac{1}{e} \] Hence, this gives maximum.

Step 5: Find maximum value. \[ y = \left(\frac{1}{x}\right)^x = \left(e\right)^{1/e} \]