Question

Cho x,y,z là các số thực dương.Chứng minh:\(\frac{x3}{y^2}\)+\(\frac{y3}{z^2}\)+\(\frac{z^3}{x^2}\)\(\ge\)\(\frac{x^2}{y}\)+\(\frac{y^2}{z}\)+\(\frac{z^2}{x}\)

Answer 1

Áp dụng BĐT Cauchy cho hai số dương :

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

\(\frac{y^3}{z^2}+y\ge2\frac{y^2}{z}\)

\(\frac{z^3}{x^2}+z\ge2\frac{z^2}{x}\)

\(\Rightarrow\frac{x^3}{y^2}+\frac{y^3}{z^2}+\frac{z^3}{x^2}\ge2\left(\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\right)-\left(x+y+z\right)\)

Mà \(\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge\frac{\left(x+y+z\right)^2}{x+y+z}=x+y+z\)

\(\Rightarrow\frac{x^3}{y^2}+\frac{y^3}{z^2}+\frac{z^3}{y^2}\ge\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\)