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  4. Solution set for (√2+√2)x+(√2-√2)x=2.2x /4 is

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Q. Solution set for $(\sqrt{2+\sqrt{2}})^x+(\sqrt{2-\sqrt{2}})^x=2.2^{x /4}$ is

Sequences and Series

Solution:

$AM \geq GM$
$\Rightarrow \frac{(2+\sqrt{2})^{x / 2}+(2-\sqrt{2})^{x / 2}}{2} \geq\left((2+\sqrt{2})^{x / 2} \cdot(2-\sqrt{2})^{x / 2}\right)^{1 / 2}$
$\Rightarrow (2+\sqrt{2})^{x / 2}+(2-\sqrt{2})^{x / 2} \geq 2(2)^{x / 4}$
Equality holds only if $(2+\sqrt{2})^{x / 2}=(2-\sqrt{2})^{x / 2}$
$\Rightarrow x=0$