# How to find the cost function for perfect complements [closed]

Imagine I got a production function like it : $$\min\{x_1, x_2\}$$

How can I find the cost function?

Hint: Think how many units of $$x_1$$ and $$x_2$$ are needed to produce $$q=1$$ in order to minimize the cost. What about producing $$q=2$$? Can you generalize it?
In order to minimize the total cost, you want to use as few units of either input as possible. Imagine you wanted to produce $$q$$ units. You would need at least $$x_1=q$$ and $$x_2=q$$ (otherwise you wouldn't be able to produce $$q$$). Therefore, to produce $$q$$, you choose precisely $$x_1(q)=q$$ and $$x_2(q)=q$$. These are the derived factor demand functions. You only need to plug them in the cost identity
\begin{align} TC(q)&=w_1 x_1(q)+w_1 x_2(q)\\ &=w_1 q +w_2 q\\ &=(w_1+w_2)q \end{align}