The minimum value of 2^ sin x +2^ cos x is - Vidyadip

MCQ

The minimum value of $2^{sin x}+2^{cos x}$ is

✓
$2^{1-\frac{1}{\sqrt{2}}}$
B
$2^{-1+\sqrt{2}}$
C
$2^{1-\sqrt{2}}$
D
$2^{-1+\frac{1}{\sqrt{2}}}$

✓

Answer

Correct option: A.

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

a
Usnign $AM \geq GM$

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

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

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

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