3. (35 points) A rm has a production function given by f (31,12) = :ri'rii, where .11 and I2 are the two inputs used to produce output 1;. Let the price the rm pays per unit of input 1 be an, and the price the rm pays per unit of input 2 be mg. The rm is planning to produce y units of output. Long-run cost-nnimization: (a) Derive the nn's conditional input demand functions and its total cost function. (b) Find the average total cost function and the marginal cost function. Docs the average total cost remain constant, increase or decrease as output increases? What do we observe that? Justify your answer. Short-run prot-maximization: In the shortrun, the rm uses 52 writs of input 2 and is unable to vary this quantity. In the short-run, the only thing that is left for the rm to choose is the amount of input 1. Let the price of the rm's output be p, and, once again, let the price the firm pays per unit of input 1 be ml, and the price the rm pays per unit of input 2 he mg. ((3) Derive the rm's short-run demand function for input 1 and its shortrun supply function. (:1) Explain what happens to the short-run profit-maximizing production plan as the output price p decreases. Praent. graphically the short-run demand function for input 1 and show on the graph what happens to the quantity demanded when p decreases. (8) Given than in] = w; = $1, '1 = $9 and 3:72 = 27, nd the iirrn's optimal level for input 1 and its optimal short-run output level. Calculate the firm's nnurinunn short-run profit level