Question

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

• A. $\{2\}$
• B. $\{0\}$
• C. $[0,2]$
• D. none of these

Answer 1

Answer: (B)

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