How to know whether Lagrange multipliers gives maximum or minimum?

Question

My book tells me that of the solutions to the Lagrange system, the smallest is the minimum of the function given the constraint and the largest is the maximum given that one actually exists. But what if we only have one point as a solution? How to know whether Lagrange multipliers gives maximum or minimum?

Answer 1

It's dictated by the nature of the problem. For example suppose you wanted to find the minimum distance from given plane to the origin of the coordinate system. This would be formulated as minimize the distance (x²+y²+z²) subject to the constraint that x,y, and z lie in the plane Ax+By+Cz-D=0. this provides the objective function and the constraint function(s)

Hope this helps.... if not let me know

Jim

Answer 2

Hi,

You usually have only one point as the solution. The classic example is: we want to find the point on the plane Ax+By+Cz=D which is nearest the origin. So we want to minimize x²+y²+z² subject to the constraint Ax+By+Cz=D

The standard Lagrange method is to form the function L= (x²+y²+z²)+λ(Ax+By+Cz-D) and solve for the point x₀,y₀,z₀ where L_x=L_y=L_z=0 which gives you 4 equations for determining x₀,y₀,z₀ and λ

You can see from the nature of the problem the result will be the point x₀,y₀,z₀ that is closest to the origin.

I hope the helps you understand this concept which is really powerful and has many applications.

Regards

Jim