For each of the following production functions: (2) Calculate the marginal rate of technical substitution (if it is defined; if it is not, indicate that that's the case), (ii) Solve for a firm's cost if the wage is $2, the rental rate on capital is $4, and the firm wishes to produce 81 units of output, (iii) Solve for the firm's costs if the wage and rental rate are the same as part (ii) but the firm wishes to produce 243 units of output, (iv) Sketch diagrams that illustrate firm's optimal choices for parts (ii) and (iii). Draw a diagram for each production function in (a) - (d). On each diagram, draw the relevant isoquants and isocost lines, indicating the optimal input demand for the target levels of output. 1. f(L, K) = LAKE 2. f (L, K) = L +3K