Question

Cho a > 0, b > 0. Chứng minh: \(\sqrt{a}\left(\sqrt{\frac{a}{b}}-1\right)\ge\sqrt{b}\left(1-\sqrt{\frac{b}{a}}\right)\)

Answer 1

\(\left\{{}\begin{matrix}\sqrt{a}=x\\\sqrt{b}=y\end{matrix}\right.\)

\(bdt\Leftrightarrow x\left(\frac{x}{y}-1\right)\ge y\left(1-\frac{y}{x}\right)\Leftrightarrow\frac{x^2}{y}-x\ge y-\frac{y^2}{x}\)

\(\Leftrightarrow\frac{x^2}{y}+\frac{y^2}{x}-x-y\ge0\)

bđt này hiển nhiên đúng theo Cauchy-Schwarz:

\(\frac{x^2}{y}+\frac{y^2}{x}\ge\frac{\left(x+y\right)^2}{x+y}=x+y\Rightarrow\frac{x^2}{y}+\frac{y^2}{x}-x-y\ge0\)

\("="\Leftrightarrow x=y\Rightarrow a=b\)