Question

Show that $2^{\sin x}+2^{\cos x}\ge 2^{1-1/\sqrt{2}}$.

Answer 1

$\frac{1}{2}[2^{\sin x}+2^{\cos x}]\ge \sqrt{2^{\sin x}.2^{\cos x}}$
$[\because AM\ge G.M]$
$\Rightarrow 2^{\sin x}+2^{\cos x}\ge 2.2^{\frac{(\sin x+\cos x)}{2}}$
Now $\sin x+\cos x=\sqrt{2}\sin (x+\frac{\pi }{4})\ge -\sqrt{2}$
$\therefore 2^{\sin x}+2^{\cos x}\ge {2.2}^{\frac{(\sin x+\cos x)}{2}}>{2.2}^{-\frac{\sqrt{2}}{2}}=2^{1-\frac{1}{\sqrt{2}}}$