Question

The minimum value of 2^sinx + 2^cosx is ______.

विकल्प

• `2^(-1 + 1/sqrt(2))`

• `2^(-1 + sqrt(2))`

• `2^(-1 - sqrt(2))`

• `2^(1 - 1/sqrt(2))`

Answer 1

The minimum value of 2^sinx + 2^cosx is `underlinebb(2^(1 - 1/sqrt(2))`.

Explanation:

`(2^(sinx) + 2^(cosx))/2 ≥ (2^(sinx + cosx))^(1/2)`  ...(∵ AM ≥ GM)

`\implies 2^(sinx) + 2^(cosx) ≥ 2.2^((sinx + cosx)/2)` 

Since, `-sqrt(2) ≤ sinx + cos x ≤ sqrt(2)`

∴ Minimum value of `2^((sinx + cosx)/2) = 2^(1/sqrt(2))` 

`\implies 2^(sinx) + 2^(cosx) ≥ 2^(1 - 1/sqrt(2))`