Solve the following

$i.$ For $0x1,$ show that $\frac{x^2}{\sqrt[\phantom{2}]{2}}\frac{x^2}{\sqrt[\phantom{2}]{1+x}}{x}^2$

$ii.$ Use the estimate $in(i)$ and integrate $itto$ prove that $\frac{1}{3\sqrt[\phantom{2}]{2}}{\displaystyle \underset{0}{\overset{1}{}}}\frac{x^2}{\sqrt[\phantom{2}]{1+x}}dx\frac{1}{3}$

iii. $By$ estimating $\frac{2x}{sinx}on$ the interval $\frac{}{6},\frac{}{2},$ prove that $\frac{2{}^2}{9}{\displaystyle \underset{\frac{}{6}}{\overset{\frac{}{2}}{}}}\frac{2x}{sinx}\frac{4{}^2}{9}$