A) \[x{{=}^{6}}\sqrt{\frac{11a}{5b}}\]
B) \[x{{=}^{6}}\sqrt{\frac{a}{2b}}\]
C) \[x=0\]
D) \[x{{=}^{6}}\sqrt{\frac{2a}{b}}\]
Correct Answer: D
Solution :
\[Condition\,\,for\,\,stable\,\,euilibrium\,\,F=\frac{dU}{dx}=0\] \[\Rightarrow \,\,\,\,\,\,-\,\frac{d}{dx}\left[ \frac{a}{{{x}^{12}}}-\frac{b}{{{x}^{6}}} \right]=0\] \[\Rightarrow \,\,\,\,\,\,\,-12a{{x}^{-}}^{13}\,\,+\,\,6b{{x}^{-7}}\,\,=\,\,0\] \[\Rightarrow \,\,\,\,\,\,\,\frac{12a}{{{x}^{13}}}\,=\,\frac{6b}{{{x}^{7}}}\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{2a}{b}={{x}^{6}}\] \[\Rightarrow \,\,\,\,\,\,\,x{{=}^{6}}\sqrt{\frac{2a}{b}}\]
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