Find minimum value of 2^sin x + 2^cos x?

Question

Find minimum value of 2^sin x + 2^cos x?

How to do this? Any particular method?

I tried finding dy/dx (with y = 2^sin x + 2 ^cos x) and setting it to zero. That gave me a relation (2^sin x)/(2^cos x) = tan x(I couldnt find out x from this). Please help?

Answer 1

oops..

dy/dx=0

sin x=cos x

x =225 degree or.625pi

minimum value=2^sin225+2^cos225

=1.225094653

Answer 2

Minimize 2^ sin x + 2 ^ cos x

By AM GM inequality

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

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

Now we need to minimize

2 ^ (sin x+ cos x)/2-1

As 2^x increases with x so we need to minimize (Sin x + cos x -2)/2 or sin x + cos x

It can be shown that

1= r sin t, 1= r cos t => t = pi/4 and r = sqrt(2)

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

it is minimum when x = -pi/4

then (sin x + cos x -2)/2

= (-sqrt(2) -2)/ 2 = (1-sqrt(2))

So minimum value = 2 ^ (1- sqrt(2))

Check:

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

The above is minimum