Using the Lagrange Multiplier find the condition of output maximization and solve the given problem.

(a) A firm that uses labour 'L' with Capital 'K' wants to determine the amount of labour and capital which maximize the production/output 'Q' with a given cost constraint 'C'; (2.5)

To Max Q=f (L, K)

C*=wL+rK

Where 'w; is the wage of labour and 'r' is the rental price of capital.

(b) Given Q = 100 K0.5 L 0.5, C* = $1,000, w= $ 30, and r = $ 40 determine the amount of labor and capital that firm should use in order to maximize output. What is the level of output?