Problem $2[12$ points$]$

Consider the preferences over bundles $of$ goods $x$ and $Y$ represented $by$ the utility function $U(x,Y)=x{Y}^{\frac{1}{3}}.$

$a{P}_x=1$ and $M=100.$ Let $P_Y$ vary and find the optimal choice $ofx$ and $Yas$ a function $of{P}_{Y}x$ and $YY?(number)[3$ points$]$

$c{P}_x=4,P_Y=2,$ find the optimal quantity for both $x$ and $Yas$ a function $ofM(formulas)[5$ points$]$

See next page for problems $3,4.$