Question

Cho các số dương a,b,c thoả mãn a > b CMR:

\(\sqrt{a+c} -\sqrt{a}<\sqrt{b+c}-\sqrt{b}\)

Answer 1

\(\sqrt{a+c}-\sqrt{a}< \sqrt{b+c}-\sqrt{b}\)

\(\Leftrightarrow\sqrt{a+c}+\sqrt{b}< \sqrt{b+c}+\sqrt{a}\)

\(\Leftrightarrow\left(\sqrt{a+c}+\sqrt{b}\right)^2< \left(\sqrt{b+c}+\sqrt{a}\right)^2\)

\(\Leftrightarrow a+b+c+2\sqrt{ab+bc}< a+b+c+2\sqrt{ab+ac}\)

\(\Leftrightarrow2\sqrt{ab+bc}< 2\sqrt{ab+ac}\Leftrightarrow\sqrt{ab+bc}< \sqrt{ab+ac}\)(đúng vs a>b) .Vậy bđt cần cm đúng