The optimization problem below indicates a utility maximization problem for

Claire. $In$ the problem, $x$ and $y$ indicate Claire's daily demand for good $1$ and $2,$

respectively, and $I>0$ indicates her daily income $(in$ dollars$).$ Which $of$ the

following statements $is$ not true?

$ma{x}_{x,y}0y^2+xy,$ subject $to,x+4y=I$

Claire's utility function $is$ quasiconcave.

For every constant I $>0,$ the associated Lagrangian has a unique stationary point.

$If$ her income $is$ sufficiently small, the problem has a corner solution.

When her daily income $is$ $$12(i.e.,$ I $=12),$ the marginal utility $of$ income $to$ Claire $is$

two units.