Explanation

(c)
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 function,
$\sin{(\phi + 2x)}$ will be minimum
$ie,\ \ \ \sin{(\phi + 2x) = - 1}$
$\therefore\ \ \ f(x) = 3^{5( - 1)} = \frac{1}{243}$