Consider a firm with the following production function employing capital and labor:

f(K, L) = 3LK +K

where the input prices are w = 6 and r = 10.

(i) Write the total cost function.

(ii) Find the optimal (cost-minimizing) mix of capital and labor the firm should employ in the long-run to produce a target output of Q = 125 while minimizing costs.

(iii) Find the optimal (cost-minimizing) mix of capital and labor the firm should employ in the long-run to produce a target output of Q = 20 while minimizing costs.

(iv) The wage rate w increases tow'=10 while the interest rate remains stable. Obtain the new optimal (cost-minimizing) mix of capital and labor the firm should employ in the long-run to produce a target output of Q = 125.

(v) How does the answers for (iv) compare to those for (ii) and (iii) and what do you learn from them?

Please solve this problem manually and correctly without using chatgbt (because it often makes calculation errors and gives the wrong solutions)