Question

chứng minh rằng,với hai số a,b thỏa mãn a>b>0 thì \(\sqrt{a}-\sqrt{b}\)<\(\sqrt{a-b}\)

Answer 1

\(\sqrt[]{a}-\sqrt[]{b}< \sqrt[]{a-b}\left(a>b>0\right)\)

\(\Leftrightarrow\left(\sqrt[]{a}-\sqrt[]{b}\right)^2< \left(\sqrt[]{a-b}\right)^2\)

\(\Leftrightarrow a+b-2\sqrt[]{ab}< a-b\)

\(\Leftrightarrow2\sqrt[]{ab}-2b>0\)

\(\Leftrightarrow2\sqrt[]{b}\left(\sqrt[]{a}-\sqrt[]{b}\right)>0\left(1\right)\)

mà \(a>b>0\)

Nên \(\left(1\right)\) luôn luôn đúng

Vậy \(\sqrt[]{a}-\sqrt[]{b}< \sqrt[]{a-b}\)