Prove that for any two numbers `x_(1) and x_(2)`
`(2e^(x_1)+e^(x_2))/(3)gte(2x_(1)+x_(2))/(3)`

Explanation

consider functionn f(x) =`e^(x)`
Graph of the function is concave upward.
Now consider two points `ax_(1),e^(x_(1))` and `B(x_(2)),e^(x_(2))` on the graph of function.
Let c Be another point which divides AB internaly in ratio 1:2
`therefore` corrdinate of point C are `(2x_(1)+x_(2))/(3),2e^(x_(1))+e^(x_(2))/(3)`
Since graph is concave upward,
`f(2x_(1)+x_(2))/(3)lt2e^(x_(1))+(e^(x_(2))/(3))`
or `(2x_(1)+x_(2))/(3)lt (e^(2x_(1))+e^(x_(2))/(3))`