Question

cho a\(\ge\)b\(\ge\)0 Chứng minh rằng \(\sqrt{a-b}\)\(\ge\)\(\sqrt{a}-\sqrt{b}\)

Answer 1

\(\sqrt{a-b}\ge\sqrt{a}-\sqrt{b}\)

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

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

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

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

\(\Leftrightarrow2\sqrt{b}\left(\sqrt{a}-\sqrt{b}\right)\ge0\)(*)

Vì \(a\ge b\Leftrightarrow\sqrt{a}\ge\sqrt{b}\Leftrightarrow\sqrt{a}-\sqrt{b}\ge0\)

Do đó (*) luôn đúng

Ta có đpcm.