A firm's production function is given by Q = 5K1/3 + 10L1/3, where K and L denote quantities of capital and labor, respectively.

1. Derive expressions (formulas) for the marginal product of each input.
2. Does more of each input increase output?
3. Does each input exhibit diminishing marginal returns? Prove.
4. Derive an expression for the marginal rate of technical substitution (MRTS) of labor for capital.
5. Suppose the price of capital, r = 1, and the price of labor, w = 1. The firm wants to produce 200 units of output. What quantities of capital and labor would minimize the costs of producing 200 units of output? Show all your work.