A) \[{{\left( \frac{1}{e} \right)}^{e}}\]
B) \[{{e}^{\frac{1}{e}}}\]
C) 1
D) 0
Correct Answer: B
Solution :
[b] \[\because Let\,y={{x}^{1/x}}\] \[logy=\frac{1}{x}logx\] Differentiating both sides, we have \[\frac{1}{y}.\frac{dy}{dx}=\frac{1}{x}.\frac{1}{x}+\left( \frac{-1}{{{x}^{2}}} \right)\log x=\frac{1}{2}(1-\log x)\] For maximum and minimum value, \[\frac{dy}{dx}=0\Rightarrow {{x}^{\frac{1}{x}-2}}.(1-\log x)=0\] If \[logx=1\] \[\Rightarrow x={{e}^{1}}=e\] \[\therefore \] Maximum value of \[y={{x}^{1/x}}={{e}^{1/e}}.\] Hence, option [b] is correct.
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