Question

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

Answer 1

${27}^{\cos \left(2x\right)}{.81}^{\sin \left(2x\right)}=3^{3\cos \left(2x\right)}{.3}^{4\sin \left(2x\right)}=3^{3\cos \left(2x\right)+4\sin \left(2x\right)}$

$\Rightarrow M=3\cos \left(2x\right)+4\sin \left(2x\right)$

The above expression reaches its maximum and minimum values when $M$ reaches its maximum and minimum values respectively.

The amplitude of $a\cos \theta +b\sin \theta$ is $A=\sqrt{a^2+b^2}$.

So, the amplitude is $\sqrt{3^2+4^2}=5$.

The minimum and maximum value of exponents is $-5$ and $5$.

The minimum value of M is clearly -5 and this is attained when

Thus the minimum value of the given expression is:

$3^{-5}=\frac{1}{243}=0.00412$