A firm wants to know how to minimize its costs for its new production plant, for any desired level of output. Suppose the production is given by y = f(x₁, x₂) = 2 x₁^1/2 x₂^1/2, where x₁ and x₂ are inputs, and the input prices are w₁ and w₂.

i. Sketch the cost minimization problem. Label your axes and graph appropriately. (1 point)

ii. Find the optimal choices of x₁ and x₂. (2 points)

iii. Find the cost function. (1 point)

iv. Now suppose input prices are given. Let the price of each input be equal to 4. In this case, what would be the firm's equation for marginal cost?