A) Increasing in \[(1,\,\,\infty )\]
B) Decreasing in \[(1,\,\,\infty )\]
C) Increasing in \[(1,\,e),\] decreasing in \[(e,\infty )\]
D) Decreasing in \[(1,\,e),\] increasing in \[(e,\infty )\]
Correct Answer: C
Solution :
Let \[y={{x}^{1/x}}\] Þ \[\log y=\frac{1}{x}\log x\] Þ \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{{{x}^{2}}}-\frac{\log x}{{{x}^{2}}}=\frac{1-\log x}{{{x}^{2}}}\] Þ \[\frac{dy}{dx}={{x}^{1/x}}\left( \frac{1-\log x}{{{x}^{2}}} \right)\] Now, \[{{x}^{1/x}}>0\] for all x and \[\frac{1-\log x}{{{x}^{2}}}>0\] in (1, e) and \[\frac{1-\log x}{{{x}^{2}}}<0\] in \[(e,\infty )\] \ \[f(x)\] is increasing in (1, e) and decreasing in \[(e,\,\infty ).\]
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