Question

Cho x²+y²+z²》1  (x,y,z>0)

Chứng minh:( x³/y)+(y³/z)+(z³/x)》1

 

Answer 1

\(\frac{x^3}{y}+xy\ge2\sqrt{\frac{x^3}{y}.xy}=2x^2\)

\(\Rightarrow\frac{x^3}{y}+\frac{y^3}{z}+\frac{z^3}{x}\ge2\left(x^2+y^2+z^2\right)-xy-yz-zx\ge2\left(x^2+y^2+z^2\right)-\left(xy+yz+zx\right)=1\)

Answer 2

\(\frac{x^3}{y}+xy\ge2\sqrt{\frac{x^3}{y}.xy}=2x^2\)

\(\frac{x^3}{y}+\frac{y^3}{z}+\frac{z^3}{x}\ge2\left(x^2+y^2+z^2\right)-\left(xy+yz+zx\right)\ge2\left(x^2+y^2+z^2\right)-\left(x^2+y^2+z^2\right)=1\)