CUET · MATHS · PYQ PAPER 2025
The maximum value of \(f(x)=\left(\frac{1}{x}\right)^x\) is
- A \(e^{-1 / e}\)
- B \(\left(\frac{1}{e}\right)^e\)
- C \(e^{1 / e}\)
- D \(\left(\frac{1}{e}\right)^{-e}\)
Answer & Solution
Correct Answer
(C) \(e^{1 / e}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\left(\frac{1}{x}\right)^x = x^{-x}\) \(\ln f(x) = -x \ln x\) \(\frac{f'(x)}{f(x)} = -(\ln x + 1)\) \(f'(x) = -f(x)(\ln x + 1)\) \(f'(x) = 0 \implies \ln x + 1 = 0 \implies \ln x = -1 \implies x = e^{-1} = \frac{1}{e}\)…
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