Find the largest interval M \subseteq \mathbb{R^ \plus{} }, such that for all a, b, c, d∈M the inequality
\sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}
holds. Does the inequality
\sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}
hold too for all a, b, c, d∈M?
( \mathbb{R^ \plus{} } denotes the set of positive reals.) inequalitiesinequalities proposed