A) \[{{e}^{1/2}}\]
B) \[{{e}^{-1/2}}\]
C) \[{{(2e)}^{-1}}\]
D) \[2{{e}^{-1/2}}\]
Correct Answer: B
Solution :
[b] Given, velocity is \[v={{x}^{2}}\log \frac{1}{x}=-{{x}^{2}}\log x\] where x is displacement. For maximum velocity, \[\frac{dv}{dx}=0\] Now, \[\frac{dv}{dx}=-{{x}^{2}}\frac{1}{x}+\log x(-2x)\] \[=-x-2x\log x\] \[\frac{dv}{dx}=0\Rightarrow -x-2x\log x=0\Rightarrow x=-2x\log x\] \[\Rightarrow \frac{-1}{2}=\log x\Rightarrow x={{e}^{-\frac{1}{2}}}\] Hence, for maximum velocity \[x={{e}^{-1/2}}\]
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