Suppose a consumer has the following utility function

$u(x_1,x_2)=2\sqrt[\phantom{2}]{x_1{x}_2}(x_1,x_2)R_{+}^2.$

Let the price vector $p>>0$ and the consumer's income $y>0.$

$($i$)$ Clearly, $u$ is continuous so that it represents a preference relation $.$ Is $$ locally insatiable? If yes, prove it$.$ If no$,$ explain the reason.

$($ii$)$ Is $u$ a concave function for all $x_1>0,x_2>0?$ If yes, prove it$.$ If no$,$ explain the reason.

$($iii$)$ Find the Marshallian demand function $d(p,y)$ and indirect utility function $v(p,y).$

$($iv$)$ Verify that $d(p,y)$ and $v(p,y)$ are homogeneous of degree $0$ in $(p,y).$

$($v$)$ Verify that the two goods are superior.