Question

Cho x<0<2, tìm GTNN của A= \(\dfrac{9x}{2-x}+\dfrac{2}{x}\)

Answer 1

Lời giải:

Ta có:

\(A=\frac{9x}{2-x}+\frac{2}{x}=\frac{-9(2-x)+18}{2-x}+\frac{2}{x}\)

\(=-9+\frac{18}{2-x}+\frac{2}{x}\)

Áp dụng BĐT Bunhiacopxky:

\(\left(\frac{18}{2-x}+\frac{2}{x}\right)(2-x+x)\geq (\sqrt{18}+\sqrt{2})^2\)

\(\Rightarrow \frac{18}{2-x}+\frac{2}{x}\geq\frac{(\sqrt{18}+\sqrt{2})^2}{2}=16\)

Do đó: \(A\geq -9+16=7=A_{\min}\)

Dấu "=" xảy ra khi \(\frac{\sqrt{18}}{2-x}=\frac{\sqrt{2}}{x}\Leftrightarrow x=\frac{1}{2}\)