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show that 2^(sin x)+2^(cos x)ge2^(1-(1)/...

show that `2^(sin x)+2^(cos x)ge2^(1-(1)/sqrt(2))`

Text Solution

Verified by Experts

Since AM of two positive quantities `ge` their GM.
`(2^(sinx)+2^(cosx))/(2)gesqrt(2^(sinx)2^(cosx))`
`=sqrt(2^(sinx)2^(cosx))`
`sqrt(2^(sqrt2sin(x+(pi)/(4))))gesqrt(2^(-sqrt(2)))`
`rArr 2^(sin x)+2^(cosx)ge2.2^((-1)/sqrt(2))=2^(1-(1)/sqrt(2))`
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