A) \[{{(e)}^{e}}\]
B) \[{{(e)}^{e}}\]
C) \[{{(e)}^{-e}}\]
D) \[{{\left( \frac{1}{e} \right)}^{e}}\]
Correct Answer: B
Solution :
\[f(x)={{\left( \frac{1}{x} \right)}^{x}}\]Þ \[f'(x)={{\left( \frac{1}{x} \right)}^{x}}\left( \log \frac{1}{x}-1 \right)\] \[f'(x)=0\Rightarrow \log \frac{1}{x}=1=\log e\Rightarrow \frac{1}{x}=e\Rightarrow x=\frac{1}{e}\] Therefore maximum value of function is \[{{e}^{1/e}}\].
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