Q: Find MC if f(L,K)=LK1/2, w=1, and v=4.
A: For a C-D production function we use RTS=w/v. In this problem RTS is 2K/L and w/v is 1/4. So the optimal condition is 8K=L. Since q=LK1/2, we know that q=(8K)K1/2 or q=8K3/2. Dividing both sides by 8 gives q/8=K3/2. Raising both sides to the 2/3 power gives (q2/3)/4=K.
Costs are wL+vK=1*(8K)+4K= 12K=12*(q2/3)/4 = 3(q2/3). MC =dTC/dq= 2q-1/3.
Additional Note: Notice that this function has increasing returns to scale and ATC=TC/q=3q-1/3, which is below MC. So ATC is falling as we would expect. You can also see that ATC is falling by taking the derivative of ATC. That is dATC/dq = -q-4/3, which is less than 0 when q > 0.