Question:
The minimum value of $2^{\sin x}+2^{\cos x}$ is :
Correct Option: , 4
Solution:
$\frac{2^{\sin x}+2^{\cos x}}{2} \geq\left(2^{\sin x+\cos x}\right)^{\frac{1}{2}} \quad(\because \mathrm{AM} \geq \mathrm{GM})$
$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2 \cdot 2^{\frac{\sin x+\cos x}{2}}$
Since, $-2 \leq \sin x+\cos x \leq \sqrt{2}$
$\therefore$ Minimum value of $2^{\frac{\sin x+\cos x}{2}}=2^{-\frac{1}{\sqrt{2}}}$
$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1-\frac{1}{\sqrt{2}}}$
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