A) \[\frac{1}{e}\]
B) \[{{e}^{1/e}}\]
C) e
D) \[\frac{1}{{{e}^{e}}}\]
Correct Answer: B
Solution :
\[y={{x}^{1/x}}\], Taking log , we have \[\log y=\frac{1}{x}\log x\] Differentiate both sides w.r.t. x \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{{{x}^{2}}}-\frac{\log x}{{{x}^{2}}}\] Þ \[\frac{dy}{dx}=\frac{1}{{{x}^{2}}}(1-\log x){{x}^{1/x}}\] For maximum, \[\frac{dy}{dx}=0\] Þ \[x=e\]; \ \[{{y}_{\max }}={{e}^{1/e}}\].
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