Use lagrange multipliers to find the point on the plane x â 2y + 3z = 6 that is closest to the point (0, 2, 4).

The distance between a point on the given plane and the point (0, 2, 4) is



but since and share critical points, we can instead consider the problem of optimizing subject to .

The Lagrangian is



with partial derivatives (set equal to 0)






Solve for :




which gives the critical point



We can confirm that this is a minimum by checking the Hessian matrix of :



is positive definite (we see its determinant and the determinants of its leading principal minors are positive), which indicates that there is a minimum at this critical point.

At this point, we get a distance from (0, 2, 4) of