On a: \((x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\)= \(1+\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+1+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}+1\) ⇾=3+\(\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}\) ① d’autre part: on a (x-y)²≥ 0 ⇾ x²+y²≥2xy et x,y>0 ⇾ \(\frac{x^{2}}{xy}+\frac{y^{2}}{xy}\)≥2 ⇾ \(\frac{x}{y}+\frac{y}{x}\)≥2 ① ⇾ 3+\(\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}\) ≥ 3+2+2+2=9
on a (x-y)²≥ 0 → x²+y²≥2xy & x,y≥0 → \(\frac{x²}{xy}+\frac{y²}{xy}≥2\) → \(\frac{x}{y}+\frac{y}{x}≥2\) ① d’autre part: on pose A= \(\frac{x+y}{z}+\frac{y+z}{x}+\frac{z+x}{y}\) A= \(\frac{x}{z}+\frac{y}{z}+\frac{y}{x}+\frac{z}{x}+\frac{z}{y}+\frac{x}{y}\) A= \((\frac{x}{z}+\frac{z}{x})+(\frac{y}{z}+\frac{z}{y})+(\frac{x}{y}+\frac{y}{x})\) ① →\(\frac{x}{z}+\frac{z}{x}≥2\) et \(\frac{y}{z}+\frac{z}{y}≥2\) et \(\frac{x}{y}+\frac{y}{x}≥2\) Donc: \(\frac{x+y}{z}+\frac{y+z}{x}+\frac{z+x}{y}≥6\)
Exercice 5:
\(x,y,z\) trois nombres réels strictement positifs et m∊IR Tel que: \(xyz = 1\) et \(\frac{2mx}{xy+x+1}+\frac{2my}{yz+y+1}+\frac{2mz}{zx+z+1}=1\)