Economics question

A rm produces two goods X and Y, with the quantity ct each sold per week being x and y respectively. The nn's prot function is given by: :r= 6x+4y+xy30 The rm also faces a labour constraint. It has 140 hours of labour available each week. Good X requires one hour of labour per unit to produce, and good Y requires three hours per unit. (a) Using the Lag'ange multiplier method and showing all the appropriate steps, determine the quantities of X and Y per week that maximise the nn's prot. Assume all items produced are sold. (10 marks) (b) Show that the second order conditions for a local maximum are satised for the solution in (a). (4 marks)