Question

[Kiệt Nguyễn]

Cho x,y,z \(\ge1\)thỏa mãn \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\).CMR:

\(\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\)

Answer 1

[Nguyễn Linh Chi]

Ta có: 

\(\left(\frac{\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}}{\sqrt{x+y+z}}\right)^2=\frac{\left(\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\right)^2}{x+y+z}\le\frac{x-1}{x}+\frac{y-1}{y}+\frac{z-1}{z}=3-\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3-2=1\)

=> \(\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\)

Dấu "=" xảy ra <=> x = y = z = 3/2

Answer 2

[Khách]

sai lớp r á bn '-'

nói vs •๖ۣۜIηεqυαℓĭтĭεʂ•ッᶦᵈᵒᶫ★T&T★