Find the point(s on the surface z2=xy 1 which are closest to the point (7, 5, 0.

We're minimizing the function



with respect to the constraint . Recall that , where is continuous, attains its extrema at the same points as . This means we can work with instead.

Using Lagrange multipliers: We have the Lagrangian



with partial derivatives



The third equation tells you that , from which you can show that are the critical points.