a: \(\left(\sqrt{a}-\sqrt{b}\right)^2\ge0\)
\(\Leftrightarrow a+b-2\sqrt{ab}\ge0\)
\(\Leftrightarrow\dfrac{a+b}{2}\ge\sqrt{ab}\)
a) (Sqrt(a) - sqrt(b))^2 >= 0
a + b - 2sqrt(ab) >= 0
a+b >= 2sqrt(ab)
(a+b)/2 >= sqrt(ab) (đpcm)
a) \(\dfrac{a+b}{2}\ge\sqrt{ab}\)
\(\Leftrightarrow a+b\ge2\sqrt{ab}\)
\(\Leftrightarrow a-2\sqrt{ab}+b\ge0\)
\(\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2\ge0\left(đúng\right)\)
Dấu "=" xảy ra \(\Leftrightarrow a=b\)
b) \(\sqrt{\dfrac{a+b}{2}}\ge\dfrac{\sqrt{a}+\sqrt{b}}{2}\)
\(\Leftrightarrow\dfrac{a+b}{2}\ge\dfrac{a+b+2\sqrt{ab}}{4}\)
\(\Leftrightarrow4a+4b\ge2a+2a+4\sqrt{ab}\)
\(\Leftrightarrow2\left(a-2\sqrt{ab}+b\right)\ge0\)
\(\Leftrightarrow2\left(\sqrt{a}-\sqrt{b}\right)^2\ge0\left(đúng\right)\)
Dấu "=" xảy ra \(\Leftrightarrow a=b\)
