The minimum value of 27^{cos 2x} cdot 81^{sin | vedclass

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Question

(A) Let $y = 27^{\cos 2x} \cdot 81^{\sin 2x}$.$y = (3^3)^{\cos 2x} \cdot (3^4)^{\sin 2x} = 3^{3 \cos 2x + 4 \sin 2x}$.For $y$ to be minimum,the expone

Vedclass · Accepted Answer

$1/243$

Vedclass · Answer

(A) Let $y = 27^{\cos 2x} \cdot 81^{\sin 2x}$.$y = (3^3)^{\cos 2x} \cdot (3^4)^{\sin 2x} = 3^{3 \cos 2x + 4 \sin 2x}$.For $y$ to be minimum,the exponent $z = 3 \cos 2x + 4 \sin 2x$ must be minimum.The expression $a \cos 	heta + b \sin 	heta$ lies in the interval $[-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}]$.Here,$a = 3$ and $b = 4$,so the range of $z$ is $[-\sqrt{3^2 + 4^2}, \sqrt{3^2 + 4^2}] = [-5, 5]$.The minimum value of $z$ is $-5$.Therefore,the minimum value of $y$ is $3^{-5} = \frac{1}{3^5} = \frac{1}{243}$.

The minimum value of $27^{\cos 2x} \cdot 81^{\sin 2x}$ is:

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