Question

CMR:\(\frac{a}{\sqrt{b}}-\sqrt{a}\ge\sqrt{b}-\frac{b}{\sqrt{a}}\)

Answer 1

\(a;b>0\)

\(\Leftrightarrow\frac{a}{\sqrt{b}}-\sqrt{b}-\left(\sqrt{a}-\frac{b}{\sqrt{a}}\right)\ge0\)

\(\Leftrightarrow\frac{a-b}{\sqrt{b}}-\frac{a-b}{\sqrt{a}}\ge0\)

\(\Leftrightarrow\left(a-b\right)\left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{ab}}\right)\ge0\)

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

Dấu "=" xảy ra khi a=b