Question

For positive reals x, y, z, Find the minimum value of $(x^{2}y+y^{2}z+z^{2}x)(x{y}^2+y{z}^2+z{x}^2)$

• A. $9x^2{y}^2{z}^2$
• B. $x^2{y}^2{z}^2$
• C. $9xyz$
• D. None of these

Answer 1

The correct option is A $9x^2{y}^2{z}^2$

$(x^{2}y+y^{2}z+z^{2}x)(y^{2}x+z^{2}y+x^{2}z)=x^3{y}^3+x^2{y}^2{z}^2+x^{4}yz+y^{4}xz+z^3{y}^3+x^2{y}^2{z}^2+x^2{y}^2{z}^2+z^{4}xy+x^3{z}^3$

$\Rightarrow x^3{y}^3+x^{4}yz+y^{4}xz+z^3{y}^3+z^{4}xy+x^3{z}^3+3x^2{y}^2{z}^2=x^3{y}^3+z^3{y}^3+x^3{z}^3+xyz(x^3+y^3+z^3)+3x^2{y}^2{z}^2$

By $AM\ge GM$ , we have $x^3{y}^3+z^3{y}^3+x^3{z}^3\ge 3x^2{y}^2{z}^2$ and $xyz(x^3+y^3+z^3)\ge 3x^2{y}^2{z}^2$

So we get $(x^{2}y+y^{2}z+z^{2}x)(y^{2}x+z^{2}y+x^{2}z)\ge 9x^2{y}^2{z}^2$