Does the output maximization level of input use exist for the following function? If yes, then what is the exact input-use level?                                                                                                            y = x1 + 0.1x12 - 0.05x13 + x2 + 0.1x22 - 0.05x23

To produce the next popular toy, a company has to pay a factory $250, 000 to set up the production line. They also have to pay $25 per item for the raw materials and labor. Write  function for the average cost to produce x items. Then describe what happens to the average cost as the factory produces a large number of toys.

The production of a certain good in terms of the amount of labour invested L and the amount of capital invested K, is given by the production function Q = 5 L²/3 K ¹/3 At this moment 512 units of labour and 1 000 units of capital are invested daily. This means that at this moment 3200 units are produced daily. a) The company receives an order for 3200 units. Give an implicit equation for the combinations (L, K) resulting in a production of 3200 units. b) Determine K'(L) for L = 100 using implicit differentiation. Interpret your result in a sentence in the context of the word problem. c) Determine the following expression and interpret your result in a sentence in the context of the word problem: 100 K'(100) K d) Set up the graph of K as a function of L using a logarithmic scale on both axes. Interpret the slope in L = 100.

The function Q = L0.3 + K0.7  is an example of _____ returns to scale of production.

3.  Suppose that the feasible region of a cost minimization linear programming problem has three corners points of (5,8), (10,5), and (4,10). If the objective function is given as: Minimize Z = 2X + Y     Which of the following represents an iso-cost line?     Select one: a.   X + 2Y  =  10   b.   X ­– Y  =  10   c.   2X ­– Y  =  10   d.   2X + Y  =  10   e. none of the other options.