Use lagrange multipliers to find the points on the cone z2 = x2 + y2 that are closest to (16, 6, 0).

Minimize subject to . The Lagrangian would be



and has partial derivatives



Setting each partial derivative to 0, we have



From the third equation, it follows that either or . In the second case, we arrive at a contradiction:



since both and must be non-negative, yet this would mean e.g. . So it must be that .

The first and second equations then tell us that




from which we obtain .

Thus the points on the cone closest to (16, 6, 0) are .

Ithink it would be 1.24 pounds