The least value of `2^(sinx)+2^(cosx)`, is

Options

A. ` 2^(1-1sqrt(2))`
B. ` 2^(1+1sqrt(2)`
C. `2^sqrt(2)`
D. 2

Answer

a

Explanation

We know that, `AM ge GM`
` :. (2^(sinx)+2^(cosx))/2 ge sqrt(2^(sin x) 2 ^(cos x))`
` rArr 2^(sinx) + 2^(cos x) le 2 sqrt(2^(sin x +cosx))`
` rArr 2 ^(sin x) + 2^(cos x) ge 2 xx2^((sin x+cosx)/2)`
But ` sin x + cos x = sqrt(2) sin (x+pi/4) ge - sqrt(2)`
` :. 2^(sin x) + 2^(cosx) ge 2^(1-(sqrt(2))/2) rArr 2^(sin x) + 2 ^(cos x) ge 2^(1-1/(sqrt(2))),AA x in R`
Hence , the minimum value is ` 2^(1-1/(sqrt(2))`.