Ingalitatea mediilor:
[tex] \frac{2ab}{a+b} \leq \sqrt{ab} \leq \frac{a+b}{2} \\ I \frac{2ab}{a+b} \leq \sqrt{ab} \\ 2ab \leq (a+b) \sqrt{ab} \\ 2 \sqrt{ab} \leq a+b \\ a+b-2 \sqrt{ab} \geq 0 \\ ( \sqrt{a}- \sqrt{b} ) ^{2} \geq 0 (A) \\
II \sqrt{ab} \leq \frac{a+b}{2} \\ 2 \sqrt{ab} \leq a+b \\ a+b-2 \sqrt{ab} \geq 0 \\ a-2 \sqrt{ab} +b \geq 0 \\ ( \sqrt{a} - \sqrt{b} ) ^{2} \geq 0(A)[/tex]
