If the boundary of the set
is a more complicated curve defined by a function
for some constant
and the first-order partial derivatives of
exist, then the method of Lagrange multipliers can prove useful for determining the extrema of
on the boundary. The method of Lagrange multipliers is introduced in
Lagrange Multipliers .
Finding absolute extrema
Use the problem-solving strategy for finding absolute extrema of a function to determine the absolute extrema of each of the following functions:
on the domain defined by
and
on the domain defined by
Using the problem-solving strategy, step
involves finding the critical points of
on its domain. Therefore, we first calculate
and
then set them each equal to zero:
Setting them equal to zero yields the system of equations
The solution to this system is
and
Therefore
is a critical point of
Calculating
gives
The next step involves finding the extrema of
on the boundary of its domain. The boundary of its domain consists of four line segments as shown in the following graph:
is the line segment connecting
and
and it can be parameterized by the equations
for
Define
This gives
Differentiating
g leads to
Therefore,
has a critical value at
which corresponds to the point
Calculating
gives the
z- value
is the line segment connecting
and
and it can be parameterized by the equations
for
Again, define
This gives
Then,
has a critical value at
which corresponds to the point
Calculating
gives the
z- value
is the line segment connecting
and
and it can be parameterized by the equations
for
Again, define
This gives
The critical value
corresponds to the point
So, calculating
gives the
z- value
is the line segment connecting
and
and it can be parameterized by the equations
for
This time,
and the critical value
correspond to the point
Calculating
gives the
z- value
We also need to find the values of
at the corners of its domain. These corners are located at
The absolute maximum value is
which occurs at
and the global minimum value is
which occurs at both
and
as shown in the following figure.
Using the problem-solving strategy, step
involves finding the critical points of
on its domain. Therefore, we first calculate
and
then set them each equal to zero:
Setting them equal to zero yields the system of equations
The solution to this system is
and
Therefore,
is a critical point of
Calculating
we get
The next step involves finding the extrema of
g on the boundary of its domain. The boundary of its domain consists of a circle of radius
centered at the origin as shown in the following graph.
The boundary of the domain of
can be parameterized using the functions
for
Define
Setting
leads to
This equation has two solutions over the interval
One is
and the other is
For the first angle,
Therefore,
and
so
is a critical point on the boundary and
For the second angle,
Therefore,
and
so
is a critical point on the boundary and
The absolute minimum of
g is
which is attained at the point
which is an interior point of
D . The absolute maximum of
g is approximately equal to 44.844, which is attained at the boundary point
These are the absolute extrema of
g on
D as shown in the following figure.