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Lagrange Multipliers

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Suppose we want to find the extreme values of subject to a constraint =.

To maximize (minimize) subject to  is to find the largest (smallest) value such that the level curve  intersects . It appears from the graph above that this happens when these curves just touch each other, that is, when they have a common tangent line. Otherwise, will be increasing (decreasing) further.

  3DView

This means that the normal lines at the point ( ) where they touch are identical. So the gradient vectors are parallel; that is,

for some scalar

 

Now let's take a look at the level curves.

 

Now, if we want to maximize subject to the constraint

=

To maximize ( minimize) subject to is to find the largest (smallest) value such that the level surface intersects .

It appears from the graph below that this happens when these surfaces just touch each other, that is, when they have a common tangent plane. Otherwise will be increasing (decreasing) further.

  3DView

This means that the normal lines at the point ( )where they touch are identical. So the gradient vectors are parallel; that is,

for some scalar .

Consider the problem of finding the maximum value on the curve of intersection of =  and  = .

3DView

Notice that the tangent vector of the curve of intersection of and is perpendicular to the normal lines of both surfaces and . So the tangent vector of at the point () is the normal vector of the plane generated by the vectors

()

and  

( )

3DView

To maximize subject to the curve of intersection of and  is to find the largest value  such that the level surface  intersects the curve . It appears from the graph below  that this happens when the level surface  just touches the curve . Otherwise will be increasing further.

  3DView

This means that the normal vector to the level surface is perpendicular to the tangent vector of the curve at the point ().

As we notice above the tangent vector of the curve is the normal vector of the plane generated by the vectors

( )

and

(

So the gradient vector of at the point () lies on the plane generated by the vectors

( )

and

( )

Thus, we have

for some scalars  and .

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