Q13/100) 2 ^ sin x + 2 ^ cos x is greater than

A> 1/2

B> 2^(1/(2^(1/2)))

C> 2^(1/2)

D> 2^(1-(1/(2^(1/2))))

Solution

by AM GM inequality we have

(a+b) /2 >= sqrt(ab)

so 2^ sin x + 2^ cos x >= 2 sqrt(2^( sinx + cos x))

sin x + cos x = sqrt(2) ( sin x cos pi/4 + cos x sin pi/4) = sqrt(2) sin (x+pi/4)

lowest value is - sqrt(2)

so 2^ sin x + 2^ cos x >= 2 sqrt(2^-(sqrt(2)) = 2^(1-(1/2^(1/2))

hence D