Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Mathematics
Rosella Castellano
Rome, University of Tor Vergata
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Criteria for Increasing/Decreasing Function
De…nition
Increasing/Decreasing Function
A function f is said to be increasing on an interval I when,
for any two numbers x1 , x2 in I ,
if x1 < x2 then f (x1 ) < f (x2 ).
A function f is said to be decreasing on an interval I when,
for any two numbers x1 , x2 in I , if x1 < x2 then
f (x1 ) > f (x2 ).
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Rosella Castellano
Criteria for Increasing/Decreasing Function
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Criteria for Increasing/Decreasing Function
Let f be di¤erentiable on the interval (a, b ).
0
If f (x ) > 0 for all x in (a, b ),then f is increasing on (a, b ).
0
If f (x ) < 0 for all x in (a, b ),then f is decreasing on (a, b ).
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
f (x ) = 18x
0
f (x ) = 18
2 3
3x
2x 2 =
2 9
Criteria for Increasing/Decreasing Function
x 2 = 2 (3 + x ) (3
Rosella Castellano
x) = 0
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Criteria for Increasing/Decreasing Function
40
30
20
10
0
y
-6
-4
-2
0
2
4
6
-10
-20
-30
-40
Figure: f (x ) = 18x
Rosella Castellano
2 3
3x
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Figure: Graph of a function f (x ) whose domain is the closed interval
[a, b ].
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
De…nition
Relative Extrema
f (x ) has a relative maximum at x = r if there is some interval
(r h, r + h) (even a very small one) for which f (r ) f (x )
for all x in (r h, r + h) for which f (x ) is de…ned.
f (x ) has a relative minimum at x = r if there is some interval
(r h, r + h) (even a very small one) for which f (r ) f (x )
for all x in (r h, r + h) for which f (x ) is de…ned.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
De…nition
Absolute Extrema
f (x ) has an absolute maximum at x = r if f (r )
every x in the domain of f .
f (x ) for
f (x ) has an absolute minimum at x = r if f (r )
every x in the domain of f .
f (x ) for
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Rosella Castellano
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
In the graph the two extrema occur at endpoints and the others at
interior points: at the points labeled stationary, the tangent lines
to the graph are horizontal, and so have slope 0.
De…nition
stationary point
0
Any time f (x ) = 0, the function f has a stationary point at x
because the rate of change of f is zero there.
The extremum that occurs at a stationary point are called
stationary extremum. In general, to …nd the exact location of
each stationary point, we need to solve the equation f 0 (x ) = 0.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
There is a relative minimum at x = q, but there is no horizontal
0
tangent there since f (q ) is not de…ned.
De…nition
singular point
When f 0 (x ) does not exist for some extrema x in the domain of f ,
we say that f has a singular extremum at x.
The points that are either stationary or singular we call collectively
the critical points of f .
Other extrema are at the endpoints of the domain. They are
(almost) always either relative maxima or relative minima.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
If f (x ) is a real-valued function, then its extrema occur among the
following types of points:
Stationary Points: f (x ) has a stationary point at x if x is in
0
the interior of the domain and f (x ) = 0. To locate stationary
0
points, set f (x ) = 0 and solve for x.
Singular Points: f (x ) has a singular point at x if x is in the
0
interior of the domain and f (x ) is not de…ned. To locate
0
singular points, …nd values of x where f (x ) is not de…ned,
but f (x ) is de…ned.
Endpoints: closed intervals contain endpoints, but open
intervals do not. If the domain of f (x ) is an open interval or
the whole real line, then there are no endpoints.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Suppose that c is a critical point of the continuous function f , and
that its derivative is de…ned for x close to, and on both sides of,
x = c. Then, determine the sign of the derivative to the left and
right of x = c.
0
If f (x ) is positive to the left of x = c and negative to the
right, then f has a maximum at x = c.
0
If f (x ) is negative to the left of x = c and positive to the
right, then f has a minimum at x = c.
0
If f (x ) has the same sign on both sides of x = c, then f has
neither a maximum nor a minimum at x = c.
Example 1,2
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Theorem
Extreme Value Theorem
If f is continuous on a closed interval [a, b ], then it will have an
absolute maximum and an absolute minimum value on that
interval. Each absolute extremum must occur at either an
endpoint or a critical point.
Therefore, the absolute maximum is the largest value of f (x ) at
the endpoints and critical points, and the absolute minimum is the
smallest value.
Once we have a candidate for an extremum of f , we …nd the
corresponding point (x, y ) using y = f (x ).
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Rosella Castellano
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Rosella Castellano
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
The extreme value theorem guarantess absolute extrema over the
interval. Clearly, the important points occurs at x = a, b, c and d,
which correspond to endpoints.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Extrema
Locating Candidates for Extrema
First Derivative Test for Extrema
Extreme Value Theorem
Finding Absolute Extrema for a function f that is continuous
on [a, b ]
1
Find the critical values of f .
2
Evaluate f (x ) at the endpoints a and b and at the critical
values in (a, b )
3
The maximum value of f is the greatest of the values found in
step 2. The minimum value of f is the least of the values
found in step 2.
Example 3
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
If s (t ) represents the position of a car at time t, then its velocity is
given by the …rst order derivative:
0
v (t ) = s (t )
But one rarely drives a car at a costant speed, hence the velocity
itself may be changing.
The rate at which the velocity is changing is the acceleration.
Because the derivative measures the rate of change, acceleration is
the derivative of velocity:
0
00
a (t ) = v (t ) = s (t ) .
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
The …rst derivative of f tells us where the graph of f is rising
0
0
[where f (x ) > 0] and where it is falling [where f (x ) < 0].
The second derivative tells in what direction the graph of f curves
or bends.
Figure: f is concave up and g is concave down
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
0
The derivative f (x ) starts small but increases as the graph
0
gets steeper. Because f (x ) is increasing, its second order
00
derivative f (x ) must be positive. The curve is concave up.
0
The derivative, g (x ) decreases as we go to the right.
0
00
Because g (x ) is decreasing, its derivative g (x ) must be
negative. The curve is concave down.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
De…nition
Let f be di¤erentiable on the interval (a, b ). Then f is said to be
0
concave up (concave down) on (a, b ) if f is increasing
(decreasing) on (a,b).
Criteria for concavity
0
Let f be di¤erentiable on the interval (a, b ):
00
if f (x ) > 0 for all x in (a, b ) then f is concave up on (a, b );
if f ”(x ) < 0 for all x in (a, b ), then f is concave down on
(a, b ).
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
De…nition
Point of in‡ection
A point in the domain of f where the graph of f changes
concavity, from concave up to concave down or vice versa, is called
a point of in‡ection.
00
To locate possible points of in‡ection, list points where f (x ) = 0
00
and also interior points where f (x ) is not de…ned. Example 4
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Acceleration
Second Derivative and Concavity
Points of In‡ection
Second Derivative Test for Relative Extrema
Suppose that the function f has a stationary point at x = c, and
00
that f (c ) exists.
00
Determine the sign of f (c ).
00
If f (c ) > 0 then f has a relative minimum at x = c.
00
If f (c ) < 0 then f has a relative maximum at x = c.
00
If f (c ) = 0 then the test is inconclusive. You need to use the
…rst derivative test to determine whether or not f has a
relative extremum at x = c.
Example 5
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
An asympote is a line that a curve approaches arbitrarily closely
(i.e in …gure a-c the dashed line x=a is an asympote.
Figure: (a) lim f (x ) = ∞; (b) lim f (x ) =
x !a +
x !a +
Rosella Castellano
∞; (c) lim f (x ) = ∞
x !a +
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
De…nition
Vertical asymptote
The line x = a is a vertical asymptote for the graph of the function
f if and only if at least one of the following is true:
lim f (x ) =
∞
lim f (x ) =
∞
x !a +
x !a
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
Example
f (x ) =
lim
x !2 +
3x 5
x 2
3x 5
x 2
= ∞. This limit is su¢ cient to conclude that x = 2 is a
vertical asympote for f (x ).
5
lim 3x
∞. This limit is su¢ cient to conclude that x = 2 is
x 2 =
x !2
a vertical asympote for f (x ).
Rule for Rationa Functions
P (x )
Suppose that f (x ) = Q (x ) where P and Q are polynomial
functions and the quozient is lowest terms. The line x = a is a
vertical asymptote for the graph of f if and only if Q (a) = 0 and
P (a) 6= 0.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
We can conclude that f (x ) increases without bound as x ! 2+
and decreases without bound as x ! 2 .
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
De…nition
Horizontal asymptote
Let f be a function. The line y=b is a horizontal asymptote for the
graph of f if and only if at least one of the following is true:
lim f (x ) = b or
x !∞
Rosella Castellano
lim f (x ) = b.
x! ∞
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Example
f (x ) = 3x
x
3x 5
5
= lim 3x
2 ; xlim
!∞ x 2
x !∞ x
3x 5
3x
lim
= lim x = 3
x! ∞ x 2
x! ∞
Rosella Castellano
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
= 3;
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Vertical Asymptotes
Horizontal Asymptote
Nonvertical asymptote
De…nition
Nonvertical asymptote
Let f be a function. The line y = mx + b is a nonvertical
asymptote for the graph of f if and only if at least one of the
following is true:
lim [f (x )
x !∞
(mx + b )] = 0 or
lim [f (x )
x! ∞
(mx + b )] = 0.
If m = 0 we repeat the de…nition of horizontal asymptote.
If m 6= 0 then y = mx + b is the equation of a nonhorizontal (and
non vertical) line with slope m that is often called oblique
asymptote.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
1
2
3
4
5
6
Identify the unknown(s). These are usually the quantities
asked for in the problem.
Identify the objective function.This is the quantity you are
asked to maximize or minimize
Identify the constraint(s). These can be equations relating
variables or inequalities expressing limitations on the values of
variables.
State the optimization problem. This will have the form
“Maximize [minimize] the objective function subject to the
constraint(s).”
Eliminate extra variables. If the objective function depends on
several variables, solve the constraint for one of the unknowns
and substitute into the objective.
Find the absolute maximum (or minimum) of the objective
function.
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
Example
Cost Minimization
For insurance purposes, a manifacturer plan to fence in a 10,800
mt2 rectangular storage area adjacent to a building by using the
buildings as one side of the enclosed area. The fencing parallel to
the building faces a highway and will cost E 3 per mt. installed,
whereas the fencing for the other two sides costs E. 2 per mt
installed. Find the amount of each type of fence so that the total
cost of the fence will be a minimum.
What is the minimum cost?
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
We label the lenght of the side parallel to the buildings as x and
the lenghts of the other two sides as y (x and y are in mt).
Along the highway the cost per foot is 3 (euro), so the cost of that
fencing is 3x. Similarly, along each of the other two sides the cost
is 2y . Hence, the total cost of fencing is:
C
= 3x + 2y + 2y
= 3x + 4y .
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
In order to di¤erentiate, we need to express C as a function of one
variable only. We can accomplish this by …nding a relationship
between x and y. We now that xy=10,800, so y= 10,800
x . Then
substituting, we get:
C
10, 800
x
43, 200
= 3x +
.
x
= 3x + 4
Rosella Castellano
Further Applications of the Derivative
Increasing/Decreasing Function
Maxima and Minima
Higher Order Derivatives
Asymptotes
Solving an Optimization Problem
From the physical nature of the problem the domain is x > 0.
x=
p
dC
=3
dx
43, 200
=0
x2
14, 400, x = 120, since x > 0.
d 2C
86, 400
=
2
dx
x3
d 2C
86, 400
=
= 0.05 > 0
2
dx (x =120 )
1203
x = 120 and y =
10,800
120
= 90.
Rosella Castellano
Further Applications of the Derivative