1.

Consider the vectors u = (-15 _25 5 V = ) and the smooth function f(I, y) such that fu(2, 1) = 0, fv(2, 1) = -1. Compute Vf(2, 1) and fw (2, 1) with w = (-3, 2).A business has $24 thousand to spend on two types of machines. Machine A costs $2 thousand per unit, and machine B costs $4 thousand per unit. Assuming that the utility as a result of buying a units of A and y units of B is determined by the function f(x, y) = vx + vy, find the numbers (r, y) which should be bought to maximize the utility. Hint: Use Lagrange multipliers]