Question

cho a\(\ge0;b\ge0\). Chứng minh

a)\(a+b\ge2\sqrt{ab}\)

b)\(\sqrt{\dfrac{a+b}{2}}\ge\dfrac{\sqrt{a}+\sqrt{2}}{2}\)

Answer 1

a) \(a+b-2\sqrt{ab}\ge0\)

<=> \(\left(\sqrt{a}+\sqrt{b}\right)^2\ge0\) (luôn đúng )

=> đpcm

b) \(\sqrt{\dfrac{a+b}{2}}\ge\dfrac{\sqrt{a}+\sqrt{b}}{2}\Leftrightarrow\sqrt{\dfrac{a+b}{2}^2}\ge\left(\dfrac{\sqrt{a}+\sqrt{b}}{2}\right)^2\)

<=> \(\dfrac{a+b}{2}\ge\dfrac{a+b+2\sqrt{ab}}{4}\)

<=> \(\dfrac{2a+2b}{4}\ge\dfrac{a+b+2\sqrt{ab}}{4}\Leftrightarrow2a+2b\ge a+b+2\sqrt{ab}\)

<=> \(2a+2b-a-b-2\sqrt{ab}\ge0\)

<=> \(a-2\sqrt{ab}+b\ge0\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2\ge0\) (luôn đúng)

=> đpcm