Question:
The minimum value of $2 \sin x+2 \cos x$ is :-
Correct Option: 1
Solution:
Usnign $\mathrm{AM} \geq \mathrm{GM}$
$\Rightarrow \frac{2^{\sin x}+2^{\cos x}}{2} \geq \sqrt{2^{\sin x} \cdot 2^{\cos x}}$
$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1+\left(\frac{\sin x+\cos x}{2}\right)}$
$\Rightarrow \min \left(2^{\sin x}+2^{\cos x}\right)=2^{1-\frac{1}{\sqrt{2}}}$