Question

Find real values of x for which ${27}^{\cos 2x}\cdot {81}^{\sin 2x}$ is minimum. Also find this minimum value.

Answer 1

$E=3^{3\cos 2x+4\sin 2x}$
Now, $3\cos 2x+4\sin 2x=r\cos (2x-\alpha )$
where $3=r\cos \alpha ,4=r\sin \alpha$
i.e., $r=5,\tan \alpha =4/3$
Its minimum values is $-r$
When $\cos (2x-\alpha )=-1\cos \pi$
or $2x-\alpha =2n\pi \pm \pi$
$\therefore x=(2n\pm 1)\frac{\pi }{2}+\frac{1}{2}{\tan }^{-1}\frac{4}{3}$
and in this case min value of E is
$3^{-r}=3^{-5}=1/243$.