Find the input combination that will minimize the cost

Question

Problem

1. A firm has the production function Q = K^0.5L^0.5 and buys input K at £12 a unit and input L at £3 a unit and has a budget of £600. Use the Lagrange method to find the input combination that will maximize output, checking that second-order conditions are satisfied by using the bordered Hessian.

2. A firm operates with the production function Q = 25K^0.5L^0.4 and buys input K at £20 a unit and input L at £8 a unit. Use the Lagrange method to find the input combination that will minimize the cost of producing 400 units of Q, using the bordered Hessian to check that second-order conditions are satisfied.

The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.