Answer each part of this question for the utility function given. Justify your answers in words and using the appropriate derivatives.

a$.U(x,Y)=\frac{Y^2}{x^2}$; Is the assumption that more is better satisfied for each good? $(6$ points$)$

b$.U(x,Y)=0\mathrm{.}5xY+2x$; Does the marginal utility of $x$ diminish, remain constant, or increase as the consumer buys more $x,$ holding $Y$ constant? $(6$ points$)$

c$.U(x,Y)=x^{\frac{4}{3}}{Y}^{\frac{4}{3}}$; Does the MRS of $x$ for $Y$ diminish, remain constant, or increase as the consumer substitutes $x$ for $Y$ to the right along an indifference curve? $(7$ points$)$

d$.U(x,Y)=x+Y^{\frac{1}{2}}$; Are the associated indifference curves downward sloping and convex to the origin $($decreasing at a decreasing rate$)?(7$ points$)$