Question

TÌm Max:

A=\(\frac{xy\sqrt{z-6}+yz\sqrt{x-2}+zx\sqrt{y-4}}{xyz}\)

Answer 1

Điều kiện:......

Áp dụng BĐT AM-GM ta có:

\(A=\frac{xy\sqrt{z-6}+yz\sqrt{x-2}+xz\sqrt{y-4}}{xyz}\)

\(=\frac{\sqrt{z-6}}{z}+\frac{\sqrt{x-2}}{x}+\frac{\sqrt{y-4}}{y}\)

\(=\frac{\sqrt{6\left(z-6\right)}}{\sqrt{6}z}+\frac{\sqrt{2\left(x-2\right)}}{\sqrt{2}x}+\frac{\sqrt{4\left(y-4\right)}}{\sqrt{4}y}\)

\(\le\frac{\frac{6+z-6}{2}}{\sqrt{6}z}+\frac{\frac{2+x-2}{2}}{\sqrt{2}x}+\frac{\frac{4+y-4}{2}}{\sqrt{4}y}\)

\(\le\frac{\frac{z}{2}}{\sqrt{6}z}+\frac{\frac{x}{2}}{\sqrt{2}x}+\frac{\frac{y}{2}}{\sqrt{4}y}=\frac{1}{2}\left(\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{4}}\right)\)

Xảy ra khi \(z=12;y=8;x=4\)