Vector x+y=(2, a+b)

Derived from the vector norm inequality |x|+|y|≥|x+y|.

√( 1+a^2)+√( 1+b^2)≥√[2^2+(a+b)^2]

Divide both sides of inequality by 2.

[√( 1+a^2)+√( 1+b^2)]/2≥√{ 1+[(a+b)/2]^2}