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Lagrange Multipliers
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.
![]()
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.
![]()
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
=
.
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
(
)
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.
![]()
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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