Prove the inequality $ \sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}$

Question

Please help me to prove the inequality $$ \sqrt{a^2 + b^2} \geq \frac{|a-b|}{\sqrt{2}}. $$

score 3 · Answer 1 · answered Nov 07 '13 at 20:25

3

Hint: Square both sides and multiply by $2$, and you'll find that this is equivalent to proving that

$$2(a^2 + b^2) \ge |a - b|^2 = (a - b)^2$$

answered Nov 07 '13 at 20:25

score 1 · Answer 2 · answered Nov 07 '13 at 20:27

1

Hint: Note that $(a-b)^2+(a+b)^2=2(a^2+b^2)$.

answered Nov 07 '13 at 20:27

André Nicolas

2 Answers2