Question

The minimum value of $\frac{X}{Y}+\sqrt{\frac{Y}{Z}}+\sqrt[3]{3\frac{Z}{X}}$ for all positive real numbers x, y, z is

• A. $3^{\frac{1}{3}}{.2}^{\frac{1}{2}}$
• B. $2^{\frac{2}{3}}{.3}^{\frac{1}{2}}$
• C. $2^{\frac{1}{2}}{.3}^{\frac{2}{3}}$
• D. $2^{\frac{1}{3}}{.3}^{\frac{1}{2}}$

Answer 1

The correct option is B $2^{\frac{2}{3}}{.3}^{\frac{1}{2}}$

$\frac{X}{Y}+\sqrt{\frac{Y}{Z}}+\sqrt[3]{3\frac{Z}{X}}$
Consider six positive real numbers $\frac{X}{Y},\frac{1}{2}\sqrt{\frac{Y}{Z}},\frac{1}{2}\sqrt{\frac{Y}{Z}},\frac{1}{3}\sqrt[3]{3\frac{z}{x}},\frac{1}{3}\sqrt[3]{3\frac{z}{x}}$
$\Rightarrow \frac{\frac{X}{Y}+\frac{1}{2}\sqrt{\frac{Y}{Z}}+\frac{1}{2}\sqrt{\frac{Y}{Z}}+\frac{1}{3}\sqrt[3]{3\frac{z}{x}}+\frac{1}{3}\sqrt[3]{3\frac{z}{x}}}{6}\ge \sqrt[6]{6\frac{X}{Y}\times \frac{1}{2}\sqrt{\frac{Y}{Z}}\times \frac{1}{2}\sqrt{\frac{Y}{Z}}\times \frac{1}{3}\sqrt[3]{3\frac{z}{x}}\times \frac{1}{3}\sqrt[3]{3\frac{z}{x}}}$
$\Rightarrow \frac{X}{Y}+\sqrt{\frac{Y}{Z}}+\sqrt[3]{3\frac{z}{z}}\ge 6\times \sqrt[6]{6\frac{X}{Y}\times \frac{Y}{Z}}\times \frac{Z}{X}\times \frac{1}{2\times 2\times 3\times 3\times 3}$
$\Rightarrow \frac{X}{Y}+\sqrt{\frac{Y}{Z}}+\sqrt[3]{3\frac{z}{z}}\ge \frac{6}{\sqrt[6]{62^2\times 3^3}}$
$\ge \frac{2\times 3}{2^{\frac{1}{3}}\times 3^{\frac{1}{2}}}$
$\ge 2^{\frac{2}{3}}\times 3^{\frac{1}{2}}$
Hence, the correct answer is option (2).