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Another extrema of sum (now happy, rmtf1111?)

Source: Romanian TST for 2019 IMO

October 1, 2019
inequalitiesalgebra

Problem Statement

Determine the largest value the expression ∑1≤i<j≤4(xi+xj)xixj \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} may achieve, as x1,x2,x3,x4 x_1,x_2,x_3,x_4 run through the non-negative real numbers, and add up to 1. 1. Find also the specific values of this numbers that make the above sum achieve the asked maximum.
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