Question

Cho \(a\ge3,b\ge4,c\ge2\) Tìm GTLN của \(A=\frac{ab\sqrt{c-2}+bc\sqrt{a-3}+ca\sqrt{b-4}}{abc}\)

Answer 1

*Sửa đề: tìm  GTNN

\(A=\frac{ab\sqrt{c-2}+bc\sqrt{a-3}+ca\sqrt{b-4}}{abc}\)

\(=\frac{\sqrt{c-2}}{c}+\frac{\sqrt{a-3}}{a}+\frac{\sqrt{b-4}}{b}\)

Áp dụng BĐT AM-GM ta có:

\(\frac{\sqrt{c-2}}{c}=\frac{\sqrt{2\left(c-2\right)}}{\sqrt{2}c}\ge\frac{\frac{2+c-2}{2}}{\sqrt{2}c}=\frac{\frac{c}{2}}{\sqrt{2}c}=\frac{1}{2\sqrt{2}}\)

TƯơng tự cho 2 BĐT còn lại ta cũng có:

\(\frac{\sqrt{a-3}}{a}\ge\frac{1}{2\sqrt{3}};\frac{\sqrt{b-4}}{b}\ge\frac{1}{2\sqrt{4}}\)

Suy ra \(A\ge\frac{1}{2}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}\right)\)