Find the minimum value of 2^("sin" x) + ...

Find the minimum value of `2^("sin" x) + 2^("cos" x)`

Text Solution

Verified by Experts

The correct Answer is:

`2^(1 - (1)/(sqrt(2))`

Using `A.M. ge G.M`., we have
`2^(sin x)+2^(cosx) ge 2sqrt(2^sinx 2^cos x)=2sqrt(2^(sinx+cosx))`
Now we know that
`sin x +cos x ge -sqrt(2)`
`rArr 2^(sinx)+2^(cos x) ge 2sqrt(2^-sqrt(2))`
Hence, the minimum value of `2^sinx +2^cosx is 2(1(1)/(sqrt(2)))`

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The minimum value of f(x)= a sin x+b cos x is

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`sqrt((a^(2)+b^(2))`

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