An electronics manufacturer has two brands that they wish to release into the market, the Xtreme Sound Radio and the You-Playr CD System. Suppose that the quantities of each brand produced by the manufacture are labelled .x and y respectively, and the market prices, labelled P; and 1": , are found to be given by 12(x)=250x R(y)=150-y and the total cost to manufacture the two products is given by I C(x,y}= 10(1: +y)+200 . (a) Write down the total prot equation for the manufacturer in terms ofx and y. (0.5 marks) (13} Find any stationary points (x , y values} for the prot equation you obtained in part (a). (1.5 marks) (c) Show that there is a single prot maximizing point, and calculate the total manufacturing costs, market prices, and prot at that point. (2 marks) Suppose now that a regulator has decided to restrict the production of the two electronics systems by the manufacturer, such that now the total quantity of sound systems produced by the manufacturer must not be greater than 30. Assuming that the manufacturer produces exactlyr 80 sounds systems in total, {d)Write down the Lagrangian needed to solve this new optimization problem . {I mark) (see next page)