The minimum value of


The minimum value of $2 \sin x+2 \cos x$ is :-


  1. $2^{1-\frac{1}{\sqrt{2}}}$

  2. $2^{-1+\sqrt{2}}$

  3. $2^{1-\sqrt{2}}$


  4. $2^{-1+\frac{1}{\sqrt{2}}}$

Correct Option: 1


Usnign $\mathrm{AM} \geq \mathrm{GM}$

$\Rightarrow \frac{2^{\sin x}+2^{\cos x}}{2} \geq \sqrt{2^{\sin x} \cdot 2^{\cos x}}$

$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1+\left(\frac{\sin x+\cos x}{2}\right)}$

$\Rightarrow \min \left(2^{\sin x}+2^{\cos x}\right)=2^{1-\frac{1}{\sqrt{2}}}$

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