If a, b are in the set of real numbers, prove that:?

Question

|ab| < or = 1/2(a^2 + b^2).

Answer 1

Consider that if a, b∈R, then (|a|-|b|)²≥0. Expanding, we have that |a|²-2|a||b|+|b|²≥0, but |a|² = a², |a||b| = |ab|, and |b|² = b², so we have a²-2|ab|+b²≥0, so a²+b²≥2|ab|, so (a²+b²)/2 ≥ |ab|, so we are done.