Use the method of lagrange multipliers to find the maximum value of f(z, y) ay for positive values of x and y, subject to the constraint x + 4y = 12. 4. suppose that p(x, y) = 2x1/3y2/3 units of a product can be made using r units of labor and y units of capital. suppose that the cost per unit of labor is $16 and the cost per unit of capital is $4. use the method of lagrange multipliers to find the amounts z and y of labor and capital, respectively, that minimize the cost of producing 32 units of product. verify that for the values of r and y that minimize cost, 16