Consider a firm with a production function f $($L$,$ K$)=3$L$1/3+24$K$1/3+$ L $+$ K$.$

$($a$)$ Calculate the marginal products of both labor and capital.

$($b$)$ Calculate the MRTS$.$

$($c$)$ Suppose w $=2$ and r $=2.$ Derive the firm$$s long$-$run cost functions cLR$($q$).$

Note: Instead of the Lagrangian method, you may directly use the two optimality

conditions: $(1)$ M RT S $=$w

r and $(2)$ f $($L$,$ K$)=$ q$.$