Using calculus, show that not all goods can be inferior. $($Hint$.$ start with the identity that $Y=p_1{q}_1{p}_2{q}_2$ dots $p_N{q}_N.)$ Note that $\frac{dY}{dY}=p_1\frac{d{q}_1}{dY}{p}_2\frac{d{q}_2}{dY}$ dots $p_N\frac{d{q}_N}{dY}.$

A$.$ Although $\frac{dY}{dY}=1,$ at least one $p_i$ can be negative, which means at least one good, good $i,$ is not inferior.

B$.$ Because $\frac{dY}{dY}=0,$ at least one $\frac{d{q}_i}{dY}$ must be positive, which means at least one good, good $i,$ is not inferior.

C$.$ Because $\frac{dY}{dY}=1,$ at least one $\frac{d{q}_i}{dY}$ must be positive, which means at least one good, good $i,$ is not inferior.

D$.$ Although $\frac{dY}{dY}=1,$ at least one $\frac{d{q}_i}{dY}$ can be negative, which means at least one good, good $i,$ is not inferior.

E$.$ Because $\frac{dY}{dY}=1,$ at least one $p_1$ must be positive, which means at least one good, good $i,$ is not inferior.