A manufacturing firm estimates that its total production of automobile batteries in thousands of units is given by f(x, y)=3x12y12, where x is the number of units of labor and y is the number of units of capital utilized. Labor costs are 70 per unit, and capital costs are 60 per unit. How many units each of labor and capital will maximize production if the firm can spend 42000 for these costs?
This problem can be translated into the following optimization problem. Find the relative extrema of the function
f(x, y)=3x12y12 subjected to the constraint 70x+60y=42000