A) \[\theta =\phi =\psi \]
B) \[\theta =\frac{\pi }{2}\]
C) \[\phi =\psi =\pi \]
D) any one of \[\theta ,\,\,\phi \,\,\psi \] is \[\frac{\pi }{3}\]
Correct Answer: A
Solution :
\[\because \] \[AM\ge GM\] \[\frac{\tan \,\theta \,\tan \,\phi \,+\tan \,\psi }{3}\ge \,{{(\tan \,\theta \,\tan \,\phi \,\tan \,\psi )}^{1/3}}\] The greatest vlaue of \[\tan \,\theta \,\tan \,\phi \,\tan \,\psi \] is \[{{\left( \frac{\tan \,\theta \,+\tan \,\phi \,+\tan \,\psi }{3} \right)}^{3}}\] In this case \[AM=GM\] and numbers will be equal. Hence, the greatest value \[=\frac{1}{\sqrt{3}}\times \frac{1}{\sqrt{3}}\times \frac{1}{\sqrt{3}}=\frac{1}{3\sqrt{3}}\] at \[\theta =\phi =\psi =\frac{\pi }{6}\]
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