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

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Question

(C) Let $f(x) = 27^{\cos 2x} 81^{\sin 2x} = (3^3)^{\cos 2x} (3^4)^{\sin 2x} = 3^{3 \cos 2x + 4 \sin 2x}$.To find the minimum value,we analyze the expo

Vedclass · Accepted Answer

$\frac{1}{243}$

Vedclass · Answer

(C) Let $f(x) = 27^{\cos 2x} 81^{\sin 2x} = (3^3)^{\cos 2x} (3^4)^{\sin 2x} = 3^{3 \cos 2x + 4 \sin 2x}$.To find the minimum value,we analyze the exponent $g(x) = 3 \cos 2x + 4 \sin 2x$.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 $3 \cos 2x + 4 \sin 2x$ is $[-\sqrt{3^2 + 4^2}, \sqrt{3^2 + 4^2}] = [-5, 5]$.The minimum value of the exponent is $-5$.Therefore,the minimum value of $f(x) = 3^{-5} = \frac{1}{3^5} = \frac{1}{243}$.

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

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