about to only mathematics

Options

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

Answer

B

Explanation

Using `A.M.geG.M.,` we have
`(2^(sinx)+2^(cosx))/(2)ge(2^(sinx)xx2^(cosx))^(1//2)`
`implies" "2^(sinx)+2^(cosx)ge2{2^((1)/(2)(sinx+cosx))}`
`implies" "2^(sinx)+2^(cosx)ge2^(1+(1)/(2)(sinx+cosx))" "...(i)`
Now,
`-sqrt(2)lesinx+cosxlesqrt(2)" for all "x`
`implies" "-(1)/(sqrt(2))le(1)/(2)(sinx+cosx)le(1)/(sqrt(2))`
`implies" "1-(1)/(sqrt(2))le1+(1)/(2)(sinx+cosx)le1+(1)/(sqrt(2))`
`implies" "2^(1-(1)/(sqrt(2)))le2^(1+(1)/(2)(sinx+cosx))le2^(1+(1)/(sqrt(2)))" "...(ii)`
From (i) and (ii), we get
`2^(sinx)+2^(cosx)ge2^(1-(1)/(sqrt(2)))" for all "x in R.`
Hence, the least value of `2^(sinx)+2^(cosx)" is "2^(1-(1)/(sqrt(2)))`