A) \[-5\]
B) \[\frac{1}{5}\]
C) \[\frac{1}{243}\]
D) \[\frac{1}{27}\]
Correct Answer: C
Solution :
Let \[f(x)={{27}^{\cos \,2x}}\,{{81}^{\sin \,2x}}={{3}^{3\cos \,2x+4\,\sin \,2x}}\] \[{{=}_{3}}5\left( \frac{3}{5}\cos 2x+\frac{4}{5}\sin \,2x \right)\] Let \[\frac{3}{5}=\sin \phi \,\,\,\,\,\Rightarrow \,\,\,\,\frac{4}{5}=\cos \phi \] then \[f(x)={{3}^{5(\sin \phi \,\,\cos \,2x+\cos \phi \,\sin 2x)}}\] \[{{3}^{5(\sin (\phi +2x))}}\] For minimum value of given fiction, \[\sin (\phi +2x)\] will be minimum, Ie, \[\sin (\phi +2x)=-1\] \[\therefore \] \[f(x)={{3}^{5(-1)}}=\frac{1}{243}\]
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