Consider a firm which produces according to technology $f(x_1,x_2)=x_1^{\frac{1}{3}}{x}_2^{\frac{2}{3}}.(*$From Tutorial $7)$

$($a$)$ Show formally whether this production function exhibits increasing, constant, or decreasing returns to scale.

$($Do not just add the powers, look at Section $19\mathrm{.}10$ of the textbook and follow the method outlined.$)$

$($b$)$ Show whether average cost is is increasing, decreasing or constant.

$($c$)$ Repeat parts $($a$)$ and $($b$)$ if the firm's technology changes to $f(x_1,x_2)=x_1{x}_2.$