Question

Cho x ≥ 1; y ≥ 2; z ≥ 3 và \(M=\dfrac{yz\sqrt{x-1}+xz\sqrt{y-2}+xy\sqrt{z-3}}{xyz}\)

Chứng minh M ≤ \(\dfrac{1}{2}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}\right)\)

 

 

Answer 1

Ta có :

\(M=\frac{yz\sqrt{x-1}+xz\sqrt{y-2}+xy\sqrt{z-3}}{xyz}\)

\(\Rightarrow M=\frac{\sqrt{x-1}}{x}+\frac{\sqrt{y-2}}{y}+\frac{\sqrt{z-3}}{z}\)

\(\Rightarrow M=\frac{2\sqrt{x-1}}{2x}+\frac{2\sqrt{y-2}.\sqrt{2}}{2y.\sqrt{2}}+\frac{2\sqrt{z-3}.\sqrt{3}}{2z.\sqrt{3}}\)

\(\Rightarrow M\le\frac{x-1+1}{2x}+\frac{y-2+2}{2y.\sqrt{2}}+\frac{z-3+3}{2z.\sqrt{3}}\)(    Áp dụng BĐT \(2xy\le x^2+y^2\))

\(\Rightarrow M\le\frac{x}{2x}+\frac{y}{2y.\sqrt{2}}+\frac{z}{2z.\sqrt{3}}\)

\(\Rightarrow M\le\frac{1}{2}+\frac{1}{2.\sqrt{2}}+\frac{1}{2.\sqrt{3}}=\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}\right)\)