A firm produces a good using two raw materials, X and Y , and the corresponding costs per unit of these raw materials are $C_x$ and $C_y$ , respectively. The amount of its good that a firm can produce using x units of X and y of Y is $\sqrt{x}+ \sqrt{y}$.
Use the Lagrange multiplier method to find the minimum combined cost, C, of raw materials X and Y which will enable the firm to produce a total amount q of its good.
Find the value, b, of the Lagrange multiplier corresponding to the optimising values of x and y. Show that b=∂C/∂q
I could not seem to find the constraint for the equation, can anyone enlighten me?I could only get the equation up till this point and I am not sure if I am on the right track. $$ \sqrt{x} + \sqrt{y} - b(x C_x + y C_y -q) = 0 $$
It sounds like you are looking to optimize the cost $C(x,y) = C_x x + C_y y$ subject to matching the total goods constraint $$\tag{*} q = \sqrt{x} + \sqrt{y}.$$
In that case, the Lagrangian is $$ \mathcal{L}(x,y) = C_x x + C_y y - b \left(\sqrt{x} + \sqrt{y} - q\right) $$ and you want to enforce $(*)$ as well.