# How to find the "cost function" given the production function *as well as* the cost per unit produced and the fixed costs?

I'm working on the following homework problem, transcribed verbatim:

A firm has a production function defined as $y = 8L^{1/4}K^{3/4}$. The firm faces costs of \$20 wage, \$60 rental rate of capital, \$2 per unit produced, and a \$42 fixed cost. Find the cost function, and average total cost, average variable cost, and marginal cost functions.

I'm confused on two levels.

(1) Does "cost function" mean short-term cost function, holding capital constant, or does it mean long-term cost function, allowing both inputs to vary? How can the professor possibly be so clueless as to fail to realize that the question is ambiguous as written?

(2) If the firm faces costs of "\$2 per unit produced and a \$42 fixed cost," shouldn't the cost function simply be $c(y)=2y+42$? But this answer contradicts the solution I get when I use the Lagrangian approach.

According to the minimization problem, the cost function $c(y)$ is the minimum value of $20L+60K$ subject to the constraint that $y=8L^{1/4}K^{3/4}$. The Lagrangian condition says

$$20=2\lambda (K/L)^{3/4}$$ $$60=6\lambda(L/K)^{1/4}$$

so

$$10(L/K)^{3/4}=10(K/L)^{1/4} \implies L=K$$

Therefore $y=8K$ and thus the cost function is $c(y)=80K=10y$. This is not the same thing as $c(y)=2y+42$. What is going on here?