Question

Cho \(B=\dfrac{x-3\sqrt{x}+2}{\sqrt{x}}\)

Tìm GTNN của \(N=\dfrac{B}{\sqrt{x}}\)

Answer 1

\(N=\frac{B}{\sqrt{x}}=\frac{x-3\sqrt{x}+2}{\sqrt{x}}\cdot\frac{1}{\sqrt{x}}=\frac{x-3\sqrt{x}+2}{x}\)

\(N=1-\frac{3\sqrt{x}}{x}+\frac{2}{x}\)

\(N=2\left(\frac{1}{x}-\frac{3}{2\sqrt{x}}+\frac{1}{2}\right)\)

\(N=2\left[\left(\frac{1}{\sqrt{x}}\right)^2-2\cdot\frac{1}{\sqrt{x}}\cdot\frac{3}{4}+\frac{9}{16}-\frac{1}{16}\right]\)

\(N=2\left(\frac{1}{\sqrt{x}}-\frac{3}{4}\right)^2-\frac{1}{8}\) \(\ge-\frac{1}{8}\forall x\)

\(N=-\frac{1}{8}\) \(\Leftrightarrow\frac{1}{\sqrt{x}}=\frac{3}{4}\)\(\Leftrightarrow x=\frac{16}{9}\)

Vậy Min N \(=-\frac{1}{8}\Leftrightarrow x=\frac{16}{9}\)