The consumer has the following utility function for goods x and $y$:$U(x,y)=x^3{y}^2.$ Let the price of x be given by $p_x,$ the price of y be given by $p_y$ and income be given by I.

$($a$)$ Derive the consumer's demand function for good x $,$ given this utility function

$2$

and budget constraint, using the Lagrange Multiplier Method.

$($b$)$ Is good $x$ normal or inferior? How can you tell?

For parts c$)$ and d$),$ let the price of x be $2.$

c$)$ Draw the consumer's Engel curve for good x $($make sure to label axes as is the norm with Engel curves$).$

d$)$ If his income is equal to $20,$ how much of good $x$ will he demand?