Analyzing Multivariable Change:
Optimization
Chapter 8.1
Extreme Points and Saddle Points
8.1- Extreme Points and Saddle Points
• The optimization techniques for functions with a
single input variable readily generalize to
multivariable functions.
• In the same way that derivatives play an
important role in determining critical points of a
function with a single input variable, partial
derivatives are used for locating critical points of
multivariable functions.
• Critical points of functions of two input variables
include maxima, minima, and saddle points.
• Optimization techniques for functions with two
input variables provides a way to fit models to
data.
Relative Extrema
• Three-dimensional functions are similar to
single-variable functions in that they may
contain relative extreme points
Relative Extrema on Contour Graphs
•
•
•
relative extreme points lie in the center of a group of concentric simple closed contours.
Contour curve levels increase as they approach a maximum point and decrease as they approach a
minimum point.
Saddle Points on Contour Graphs
Absolute Extrema
• For a relative maximum of a multivariable function, if there
are no output values greater than that relative maximum,
the relative maximum is also the absolute maximum.
Figure 8.20 shows a graph that contains relative maxima at
points A and B and an absolute maximum at point B.
• Similarly, for a relative minimum of a multivariable
function, if there are no output values less than that
relative minimum, that relative minimum is the absolute
minimum
Absolute Extrema on Tables and
Graphs
• Contour curves on tables and graphs are useful in locating
absolute extrema on tables and graphs of continuous
three-dimensional functions because absolute extrema can
be located only within concentric closed contour curves or
on terminal edges of a table or graph.
– Visually locate all relative extrema. Sketch contour curves if they
are not already present.
– Relative extrema lie within concentric closed contour curves.
– Locate any candidates on terminal edges, if any terminal edges
exist, and consider end behavior beyond non-terminal edges.
– Compare relative extrema, candidates on terminal edges, and
limits of end behavior beyond non-terminal edges.
Problem 4,6, 10
Analyzing Multivariable Change:
Optimization
Chapter 8.2
Multivariable Optimization
Multivariable Optimization
• Relative extreme points on the graph of a
–
–
–
–
continuous,
differentiable,
single-variable function
occur at points where the tangent line is horizontal.
• Derivatives are used to locate these points, whereas second
derivatives are used to identify them as maxima or minima.
• Similarly, for functions with two input variables, critical
points (relative extrema and saddle points) occur at points
where the two cross-sectional tangent lines are contained
in a plane that is horizontal (has the same output for all
input points).
When Partial Rates of Change are Zero
• A relative maximum (minimum) point (x, y) on
a function f with input is defined as a point (x0,
y0, f(x0, y0)) with output value greater than
(less than) the output values of all
surrounding points.
Relative maximum (minimum) point
• the function f is at its maximum or
Minimum, when both partial
derivatives are zero:
– Fx = 0
– Fy = 0
Saddle Points
• It can also be illustrated that
for saddle points, both partial
derivatives are zero.
Critical Points of Functions
Determinant Test
• The determinant of the second partials matrix, evaluated at a point
where fx = fy = 0 , can be used to test whether the critical point is
–
–
–
–
a relative maximum,
a relative minimum,
or a saddle point.
This test is known as the Determinant Test.
Problem 2, 4, 10, 12, 16, 20
Analyzing Multivariable Change:
Optimization
Chapter 8.3
Optimization under Constraints
Optimization under Constraints
• In many optimization applications, the context
dictates constraints on what input values can be used.
• For example, when trying to optimize revenue made
from the sales of a product, the producer is
constrained by the total amount of product the
company is able to produce.
• When a manufacturer is trying to maximize
production, budget constraints must be considered.
Constraints from a Graphical
Perspective
• Constrained optimization refers to the process of
determining the maximum (or minimum) output value
of a multivariable function when there are restrictions
placed on what input values can be used.
• The constrained optimum is the greatest (or least)
output value of f corresponding to a point on the
constraint curve.
• The input of the constrained optimal point is the point
where the constraint curve g touches a contour curve
of the constrained function at only one point without
passing through that contour curve.
• The constraint curve remains completely on one side of
the optimal contour curve except at the optimal point.
Constrained Optimal Points
• Determining constrained optimal points involves the
calculation of slopes of tangent lines.
• The slope of the tangent line at a constrained optimal point
can be calculated using
– either the multivariable function for the constraint function g.
– a contour curve of the function f at the optimal point (x0, y0,
m).
It can be shown using calculus and algebra that the slopes of these two tangent lines are
the same.
Numerical Verification of a
Constrained Optimum
• Because not all solutions to the Lagrange system have
to be optimal points of the constrained problem, it is a
good idea to verify that the solution found is really an
optimum.
• Verification can be done graphically, but it is often
easier, faster, and more accurate to verify the
optimum numerically.
• To verify numerically that a solution to the Lagrange
system is the optimal point for the constrained system,
check the output value of the solution point with the
output value of two close points on the constrained
function.
Interpretation of the Lagrange
Multiplier  (lambda)
• If the level of the constraint function g changes
from c to c + c, it will determine a different
extreme point because adding c to the
constraint equation shifts the constraint curve.
• Constraint: g(x, y) = c + c
• F(x, y) = m + m
• the Lagrange multiplier  is the rate of change of
the extreme value with respect to the constraint
level:
• The units of output of lambda are
– (output units of f ) per (output units of g)
Problem 6, 8, 10, 16, 22, 24