9. Let $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$ and $\mathrm { t } _ { 1 } = ( \tan \theta ) ^ { \tan \theta } , \mathrm { t } _ { 2 } = ( \tan \theta ) ^ { \cot \theta } , \mathrm { t } _ { 3 } = ( \cot \theta ) ^ { \tan \theta }$ and $\mathrm { t } _ { 4 } = ( \cot \theta ) ^ { \cot \theta }$, then\\
(A) $\mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 4 }$\\
(B) $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$\\
(C) $\mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 4 }$\\
(D) $\mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 4 }$

\section*{Sol. (B)}
Given $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$, then $\tan \theta < 1$ and $\cot \theta > 1$.\\
Let $\tan \theta = 1 - \lambda _ { 1 }$ and $\cot \theta = 1 + \lambda _ { 2 }$ where $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are very small and positive.\\
then $\mathrm { t } _ { 1 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 - \lambda _ { 1 } } , \mathrm { t } _ { 2 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 + \lambda _ { 2 } }$

$$t _ { 3 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 - \lambda _ { 1 } } \text { and } t _ { 4 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 + \lambda _ { 2 } }$$

Hence $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$.\\