A constrained optimization problem is a problem of the form
maximize (or minimize) the function subject
to the condition
.
In some cases one can solve for as a function of
and
then find the extrema of a one variable function.
That is, if the equation is equivalent to
,
then we may set
and then find the values
for which
achieves an extremum. The extrema of
are
at
.
Find the extrema of
subject to
.
We solve
. Set
.
Differentiating we have
.
Setting
, we must solve
, or
. Differentiating again,
so that
which shows that
is a relative minimum of
and
is a relative minimum of
subject to
.
Find the extrema of
subject to
.
Using the quadratic formula, we find
Substituting the above expression for in
we
must find the extrema of
and
and
Setting (respectively,
) we find
in each case. So the potential extrema are
and
.
Evaluating at , we see that
so that
is a
relative minimum and as
,
is a relative maximum.
(even though
!)
If is a (sufficiently smooth) function in two
variables and
is another function in two variables,
and we define
, and
is a relative extremum of
subject to
, then
there is some value
such that
.
Find the extrema of the function
subject to the constraint
.
Set
. Then
Setting these equal to zero, we see from the third equation
that , and from the first equation that
, so that from the second equation
implying that
. From the third equation, we obtain
.
Find the potential extrema of
the function
subject
to the constraint that
.
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(1) |
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(2) |
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(3) |
Multiplying the first line by and the second by
we obtain:
Subtracting, we have
As
, we conclude that
. Substituting,
we have
.
So the potential extrema are at or
.