3. Graphing and Optimization
3-1 First Derivative and Graphs
Increasing and Decreasing Functions
f  x   x 3  3x
f  x  
f  x 
f x
Graph of f  x 
+
increasing
rising
decreasing
falling
Given the function f  x   8 x  x 2 ,
What values of x correspond to horizontal tangents?
Where is f  x increasing? Decreasing?
Graph f  x ; add horizontal tangents.
Critical Values
Definition. The values of x in the domain of f
where f  x   0 or where f  x  does not exist
are called the critical values of f.
Find the critical values of f  x   1  x3 , and
intervals where it is increasing/decreasing.
Find the critical values of f  x   1  x 1 3 , and
intervals where it is increasing/decreasing.
Find the critical values of f  x   1
, and
x2
intervals where it is increasing/decreasing.
Find the critical values of f  x   8 ln x  x 2 , and
intervals where it is increasing/decreasing.
Local Extrema
Use the graph to find intervals where f is
increasing/decreasing, local max/min.
Theorem
If f is continuous on the interval (a, b), c is a
number in (a, b), and f (c) is a local extremum,
then either f (c) = 0 or f (c) does not exist. That
is, c is a critical point. (A critical value is not
necessarily a local extremum.)
First Derivative Test
Let c be a critical value of f [f (c) is defined, and
either f (c) = 0 or f (c) is not defined.]
Construct a sign chart for f (x) close to and on
either side of c.
For f  x   x3  6 x 2  9 x  1,
Find the critical values of f.
Find local max/min.
Graph.
Theorem 3.
If f  x   an x n  an1x n1    a1x  a0 , an  0 ,
is an nth-degree polynomial, then f has at most
n x-intercepts and at most (n – 1) local extrema.
Over the past few decades, The US has exported
more ag products than it imported. Graph
approximates rate of change of balance of trade.
Graph of total revenue R  x  from sale of bookcases
Describe; graph R x .
3-2 Second Derivative and Graphs
Using Concavity as a Graphing Tool
Consider f  x   x 2 and g  x   x for 0  x   .
f  x   2 x  0
1
g  x  
0
2 x
The term concave upward (or simply concave
up) is used to describe a portion of a graph that
opens upward. Concave downward is used to
describe a portion of a graph that opens
downward.
Concavity
The graph of a function f is concave upward
on the interval (a,b) if f (x) is increasing on
(a,b), and is concave downward on the interval
(a,b) if f (x) is decreasing on (a,b).
For y = f (x), the second derivative of f,
d2y
d
provided it exists, is f  x  
f  x , or 2 or
dx
dx
y x .
For f  x   x 2 , find f  x .
For f  x   x , find f  x .
Discuss concavity upward/downward for f  x   e x
Discuss concavity upward/downward for g  x   ln x
Discuss concavity upward/downward for h x   x 3
Inflection Points
An inflection point is a point on the graph
where the concavity changes from upward to
downward or downward to upward.
Theorem 1
If y  f  x  is continuous on a, b  and has an
inflection point at x = c, then either f (c) = 0 or
f (c) does not exist.
Find the inflection points for
f  x   x3  6 x 2  9 x  1
Find the inflection points for
f  x   ln x 2  4 x  5


Analyzing graphs
Curve Sketching
Graphing Strategy
 Step 1. Analyze f  x .
Find the domain and the intercepts. The x
intercepts are the solutions to f  x   0, and
the y intercept is f 0.
 Step 2. Analyze f  x .
Find the partition points critical values of f  x .
Construct a sign chart for f  x , determine the
intervals where f is increasing and decreasing,
and find local maxima and minima.
 Step 3. Analyze f  x .
Find the partition numbers of f  x .
Construct a sign chart for f  x , determine
the intervals where the graph of f is
concave upward and concave downward,
and find inflection points.
 Step 4. Sketch the graph of f.
Locate intercepts, local maxima and minima,
and inflection points. Sketch in what you
know from steps 1-3. Plot additional points
as needed and complete the sketch.
Follow the graphing strategy and analyze the
function f  x   x 4  2x3 .
Step 1. Analyze f  x .
Step 2. Analyze f  x .
Step 3. Analyze f  x .
Step 4. Sketch the graph of f.
Follow the graphing strategy and analyze the
function f  x   3x5 3  20 x .
Step 1. Analyze f  x .
Step 2. Analyze f  x .
Step 3. Analyze f  x .
Step 4. Sketch the graph of f.
Point of Diminishing Returns
If a company decides to increase spending on
advertising, they would expect sales to increase.
At first, sales will increase at an increasing rate
and then increase at a decreasing rate. The
value of x where the rate of change of sales
changes from increasing to decreasing is called
the point of diminishing returns. This is also
the point where the rate of change has a
maximum value. Money spent after this point
may increase sales, but at a lower rate.
A discount appliance store is selling 200 largescreen TV sets monthly. If the store invests $x
thousand in advertising, the ad company
estimates that sales will increase to
N  x   3 x 3  0.25 x 4  200
0 x9
When is the rate of change of sales increasing
and when is it decreasing? What is the point of
diminishing returns and the maximum rate of
change of sales? Graph N and N.
(6-2 Second-Order Partial Derivatives)
z  f  x, y   x 4 y 7
z
z
 f x  f x  x, y  
 f y  f y  x, y  
x
y
f xx
 2 z   z 
 f xx  x, y   2    
x  x 
x
f xy
 2 z   z 
 f xy  x, y  
  
yx y  x 
f yx
 2 z   z 
 f yx  x, y  
  
xy x  y 
f yy
 2 z   z 
 f yy  x, y   2    
y  y 
y
Let z  f  x, y   3x 2  2 xy 3  1
 2z
 2z
Find
and
.
xy
yx
 2z
Find 2
x
Find f yx 2,1
3- 4 Curve Sketching Techniques
When we summarized the graphing strategy in a
previous section, we omitted one very important
topic: asymptotes.
Modifying the Graphing Strategy
Step 1. Analyze f  x .
• Find the domain of f.
• Find the intercepts.
• Find asymptotes
Step 2. Analyze f  x .
• Find the partition numbers and critical
values of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where f is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze f  x .
• Find the partition numbers of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where the graph of f
is concave upward and concave downward.
• Find inflection points.
Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
Using the Graphing Strategy
Analyze and graph f  x    x  1  x  2
Step 1. Analyze f  x .
• Find the domain of f.
• Find the intercepts.
• Find asymptotes
Step 2. Analyze f  x .
f  x  
• Find the partition numbers and critical
values of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where f is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze f  x .
f  x  
• Find the partition numbers of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where the graph of f
is concave upward and concave downward.
• Find inflection points.
Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
Using the Graphing Strategy
2x 1
Analyze and graph g  x   2
x
Step 1. Analyze g  x .
• Find the domain of g.
• Find the intercepts.
• Find asymptotes
Step 2. Analyze g x .
g  x  
• Find the partition numbers and critical
values of g (x).
• Construct a sign chart for g (x).
• Determine the intervals where g is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze g x .
g x  
• Find the partition numbers of g (x).
• Construct a sign chart for g (x).
• Determine the intervals where the graph of g
is concave upward and concave downward.
• Find inflection points.
Step 4. Sketch the graph of g.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
Using the Graphing Strategy
Analyze and graph f  x   xe x
Step 1. Analyze f  x .
• Find the domain of f.
• Find the intercepts.
• Find asymptotes
Step 2. Analyze f  x .
f  x  
• Find the partition numbers and critical
values of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where f is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze f  x .
f  x  
• Find the partition numbers of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where the graph of f
is concave upward and concave downward.
• Find inflection points.
Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
Using the Graphing Strategy
Analyze and graph f  x   x 2 ln x  0.5 x 2
Step 1. Analyze f  x .
• Find the domain of f.
• Find the intercepts.
• Find asymptotes
Step 2. Analyze f  x .
f  x  
• Find the partition numbers and critical
values of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where f is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze f  x .
f  x  
• Find the partition numbers of f (x).
• Construct a sign chart for f (x).
• Determine the intervals where the graph of f
is concave upward and concave downward.
• Find inflection points.
Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
Modeling average cost
Let C  x   5,000  0.5 x 2  C  x  
Step 1. Analyze C  x .
• Find the domain of C  x .
• Find the intercepts.
• Find asymptotes
Step 2. Analyze C  x .
C  x  
• Find the partition numbers and critical
values of C  x .
• Construct a sign chart for C  x .
• Determine the intervals where C  x  is
increasing and decreasing
• Find local maxima and minima
Step 3. Analyze C  x .
C  x  
• Find the partition numbers of C  x .
• Construct a sign chart for C  x .
• Determine the intervals where the graph of
C  x  is concave up and concave down.
• Find inflection points.
Step 4. Sketch the graph of f.
• Draw asymptotes and locate intercepts,
local max and min, and inflection points.
• Plot additional points as needed and
complete the sketch
3-5 Absolute Maxima and Minima
f c  is an absolute maximum of f if f c   f  x 
for all x in the domain of f.
f c  is an absolute minimum of f if f c   f  x 
for all x in the domain of f.
Theorem 1. Extreme Value Theorem
A function f that is continuous on a closed
interval a, b has both an absolute maximum
value and an absolute minimum value on that
interval.
Theorem 2. Absolute extrema (if they exist)
must always occur at critical values of the
derivative, or at end points.
Finding Absolute Extrema on a closed interval
Step 1. Check to make sure f is continuous over a, b.
Step 2. Find the critical values in the interval a, b.
Step 3. Evaluate f at the end points a and b and
at the critical values found in step 2.
Step 4. The absolute maximum on [a, b] is the
largest of the values found in step 3.
Step 5. The absolute minimum on [a, b] is the
smallest of the values found in step 3.
Find the absolute minimum and maximum
values of f  x   x3  3x 2  9 x  7 on  6,4.
on  4,2.
on  2,2.
Second Derivative and Extrema
Suppose f c   0 and f c   0
Suppose f c   0 and f c   0
f c  f c  graph of f is
f c 
0
+
concave up
local minimum
0
concave down local maximum
0
0
?
test does not apply
Find local max/min for f  x   x3  6 x 2  9 x  1
Find local max/min for f  x   xe0.2 x
x6
Find local max/min for f  x    4 x 5  25 x 4
6
Theorem 3
Second Derivative Test for Absolute Extremum
Let f be continuous on an interval I with only
one critical value c in I.
If f c   0 and f c   0 , then f c  is the
absolute minimum of f on I.
If f c   0 and f c   0, then f c  is the
absolute maximum of f on I.
4
Find the absolute minimum of f  x   x  on
x
the open interval 0,  
Find the absolute minimum of f  x   ln x 2  3 ln x
on the open interval 0,  
3- 6 Optimization
A homeowner has $320 to spend on fencing for
his garden. 3 sides of the fence will be with
wire at a cost of $2/foot. The fourth side will be
wood at a cost of $6/foot. Find the dimensions
and the area of the largest garden that can be
enclosed for $320.
Strategy for Solving Optimization Problems
1.Introduce variables, look for relationships
among these variables, and construct a math
model of the form:
Maximize (minimize) f (x) on the interval I.
2.Find the critical values of f (x).
3.Find the maximum (minimum) value of f
(x) on the interval I (last section).
4.Use the solution to the mathematical model
to answer all the questions asked in the
problem.
The homeowner judges that 800 ft² is too small,
and decides to increase area to 1,250 ft². What
is the minimum cost of this fence? What are the
dimensions of this fence?
An office supply company sells x permanent
markers per year at $p per marker. The pricedemand equation for these markers is
p  10  0.001x . What should the company
charge for the markers to maximize revenue?
The total annual cost of manufacturing x
permanent markers is C  x   5000  2 x . What
is the company’s maximum profit, what should
the company charge per marker, and how many
should be produced?
The government decides to tax the company
$2/marker produced. Taking this into account,
how many markers should the company
manufacture each week to maximize its weekly
profit? What is the maximum weekly profit?
How much should the company charge for the
markers to maximize profit?
When a management training company prices
its seminar on management techniques at $400
per person, 1,000 people will attend the seminar.
The company estimates that for each $5
reduction in price, an additional 20 people will
attend the seminar. How much should the
company charge in order to maximize its
revenue? What is the maximum revenue?
After additional analysis, the company decides
its attendance estimate was too high. The new
estimate is that only 10 additional will attend for
each $5 reduction. How much should the
company charge now in order to maximize its
revenue? What is the new maximum revenue?
Inventory Control
A multimedia company anticipates that there
will be a demand for 20,000 copies of a DVD
during the next year. It costs the company
$0.50 to store a DVD for a year. Each setup for
DVDs costs $200. How many DVDs should the
company make in each run to minimize its total
storage and setup costs?
6- 3 Maxima and Minima
z  f  x, y 
local max
local min
saddle point
f a, b  is a local maximum if there exists a
circular region about a, b  such that
f a, b  f  x, y  for all  x, y  in the region.
f a, b  is a local minimum if there exists a
circular region about a, b  such that
f a, b  f  x, y  for all  x, y  in the region.
Theorem 1
Let f a, b  be a local extremum for the
function f. If both f x and f y exist at a, b , then
f x a, b   0 and f y a, b   0 .
(saddle point)
The following theorem, using second derivatives,
gives us sufficient conditions for a critical point
to produce a local extremum or a saddle point.
Theorem 2
Assume that
1. z  f  x, y 
2. f x a, b   0 and f y a, b   0
3.All second-order partial derivatives of f
exist in some circular region containing
a, b  as a center.
4. A  f xx a, b , B  f xy a, b , C  f yy a, b 
Then
Case 1. If AC  B 2  0 and A  0 , then f a, b 
is a local maximum.
Case 2. If AC  B 2  0 and A  0 , then f a, b 
is a local minimum.
Case 3. If AC  B 2  0 , then f a, b  is a saddle
point.
Case 4. If AC  B 2  0, the test fails.
Procedure
Let z  f  x, y   x 2  y 2
Step 1. Find critical points: Find a, b  such that
f x a, b   0 and f y a, b   0 simultaneously.
Step 2. Compute A  f xx 0,0, B  f xy 0,0,
and C  f yy 0,0.
Step 3. Evaluate AC  B 2 and classify the
critical point using theorem 2.
Let f  x, y    x 2  y 2  6 x  8 y  21
Step 1. Find critical points: Find a, b  such that
f x a, b   0 and f y a, b   0 simultaneously.
Step 2. Compute A  f xx 3,4, B  f xy 3,4,
and C  f yy 3,4.
Step 3. Evaluate AC  B 2 and classify the
critical point using theorem 2.
Let f  x, y   x3  y 3  6 xy
Step 1. Find critical points: Find a, b  such that
f x a, b   0 and f y a, b   0 simultaneously.
Step 2. Compute A  f xx a, b , B  f xy a, b ,
and C  f yy a, b  for a, b  = 0,0  and 2,2 
Step 3. Evaluate AC  B 2 and classify the
critical point using theorem 2.
Suppose the surfboard company has a profit equation
P x, y   22 x 2  22 xy  11y 2  110 x  44 y  23
x is standard, y is competition, P is profit (000).
Step 1. Find critical points: Find a, b  such that
f x a, b   0 and f y a, b   0 simultaneously.
Step 2. Compute A  f xx 3,1, B  f xy 3,1,
and C  f yy 3,1.
Step 3. Evaluate AC  B 2 and classify the
critical point using theorem 2.
The packaging department of a company is
designing a rectangular box, no top, partition in
middle. Volume must be 48”². Find dimensions
that minimize required material.
Step 1. Find critical points: Find a, b  such that
f x a, b   0 and f y a, b   0 simultaneously.
Step 2. Compute A  f xx 6,4, B  f xy 6,4,
and C  f yy 6,4.
Step 3. Evaluate AC  B 2 and classify the
critical point using theorem 2.