Class 11

Math

Algebra

Sequences and Series

The minimum value of $2_{sinx}+2_{cosx}$ is

- $2_{1−21}$
- $2_{1+21}$
- $2_{2}$
- $2$

$∴22_{sinx}+2_{cosx} ≥2_{sinx}2_{cosx} $

$⇒2_{sinx}+2_{cosx}≥22_{sinx+cosx} $

$⇒2_{sinx}+2_{cosx}≥2×2_{2sinx+cosx}$

$⇒2_{sinx}+2_{cosx}≥2_{1+2sinx+cosx}$

But $sinx+cosx=2 sin(x+4π )≥−2 $

$∴2_{sinx}+2_{cosx}≥2_{1−22}$

$⇒2_{sinx}+2_{cosx}≥2_{1−21},∀x∈R$

Hence, minimum value is $2_{1−21}$