Question

Given: `x+y+z = 5` what is the value of `2^x+2^y+2^z` ?

Answer 1

`2^x+2^y+2^z in [3 * 2^(5/3), oo)`

Explanation:

Given: `x+y+z = 5` what is the value of `2^x+2^y+2^z` ?

There is insufficient information to determine a unique solution or even a finite number of solutions.

The minimum possible value occurs when `x=y=z`, namely:

`2^x+2^y+2^z = 3 * 2^(5/3) ~~ 9.5244`

In general the value is unbounded.

For example, we could put `z=5`. Then `y = -x` and:

`2^x+2^y+2^z = 2^x+2^(-x)+32 > 2^x`

Answer 2

See below.

Explanation:

We will assume that the question is to determine the minimum for

`2^x+2^y + 2^z`

given

`x + y + z = 5`

There is an inequality (arithmetic-geometric means)

https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

witc says

Given `x_1 > 0, x_2 > 0, x_3 > 0`

`(x_1+x_2+x_3)/3 ge root(3)(x_1 x_2 x_3)`

or `x_1+x_2+x_3 ge 3 root(3)(x_1 x_2 x_3)`

Making `x_1 = 2^x > 0, x_2 = 2^y > 0, x_3 = 2^z > 0` we have

`2^x+2^y+2^z ge 3 root(3)(2^(x+y+z)) = 3xx2^(5/3)`