CHAPTER

Applications of Differentiation
3
3.1 LINEAR APPROXIMATIONS AND NEWTON’S
METHOD
3.2 INDETERMINATE FORMS AND l’HÔPITAL’S RULE
3.3 MAXIMUM AND MINIMUM VALUES
3.4 INCREASING AND DECREASING FUNCTIONS
3.5 CONCAVITY AND THE SECOND DERIVATIVE TEST
3.6 OVERVIEW OF CURVE SKETCHING
3.7 OPTIMIZATION

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 CHAPTER
Applications of Differentiation
3
3.8 RELATED RATES
3.9 RATES OF CHANGE IN ECONOMICS AND THE
SCIENCES

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 3.3 MAXIMUM AND MINIMUM VALUES

DEFINITION 3.1
For a function f defined on a set S of real numbers and a
number c ∈ S,
(i) f (c) is the absolute maximum of f on S if f (c) ≥ f (x) for
all x ∈ S and

(ii) f (c) is the absolute minimum of f on S if f (c) ≤ f (x)

for all x ∈ S.

An absolute maximum or an absolute minimum is

referred to as an absolute extremum. (The plural form of
extremum is extrema.)
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 3.3 MAXIMUM AND MINIMUM VALUES

Existence of Absolute Extrema

Functions do not necessarily have absolute extrema.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.1 Absolute Maximum and Minimum

Values
(a) Locate any absolute extrema of f (x) = x2 − 9 on the
interval (−∞,∞).

(b) Locate any absolute extrema of f (x) = x2 − 9 on the

interval (−3, 3).

(c) Locate any absolute extrema of f (x) = x2 − 9 on the

interval [−3, 3].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.1 Absolute Maximum and Minimum

Values
Solution
(a)

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.1 Absolute Maximum and Minimum

Values
Solution
(b)

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.1 Absolute Maximum and Minimum

Values
Solution
(c)

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.2 A Function with No Absolute Maximum

or Minimum
Locate any absolute extrema of f (x) = 1/x,
on [−3, 0) ∪ (0, 3].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.2 A Function with No Absolute Maximum

or Minimum
Solution
f clearly fails to have either an
absolute maximum or an
absolute minimum on
[−3, 0) ∪ (0, 3].

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 3.3 MAXIMUM AND MINIMUM VALUES

THEOREM 3.1 (Extreme Value Theorem)

A continuous function f defined on a closed, bounded

interval [a, b] attains both an absolute maximum and an
absolute minimum on that interval.

(Theorem 3.1 says that continuous functions are

guaranteed to have an absolute maximum and an
absolute minimum on a closed, bounded interval.)

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.3 Finding Absolute Extrema of a

Continuous Function
Find the absolute extrema of f (x) = 1/x on the interval
[1, 3].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.3 Finding Absolute Extrema of a

Continuous Function
Solution
On the interval [1, 3], f is
continuous. Consequently, the
Extreme Value Theorem
guarantees that f has both an
absolute maximum and an
absolute minimum on [1, 3]. Judging from the graph in the
figure, it appears that f (x) reaches its maximum value of 1
at x = 1 and its minimum value of 1/3 at x = 3.
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 3.3 MAXIMUM AND MINIMUM VALUES

DEFINITION 3.2
(i) f (c) is a local maximum of f if f (c) ≥ f (x) for all x in
some open interval containing c.

(ii) f (c) is a local minimum of f if f (c) ≤ f (x) for all x in

some open interval containing c.

In either case, we call f (c) a local extremum of f .

(Local maxima and minima are sometimes referred to as

relative maxima and minima, respectively.)
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 3.3 MAXIMUM AND MINIMUM VALUES

Local Extrema

Notice that each local

extremum seems to occur
either at a point where the
tangent line is horizontal
[i.e., where f (x) = 0], at a
point where the tangent
line is vertical [where f (x)
is undefined] or at a corner [again, where f (x) is
undefined].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.4 A Function with a Zero Derivative

at a Local Maximum

Locate any local extrema for f (x) = 9 − x2 and describe the

behavior of the derivative at the local extremum.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.4 A Function with a Zero Derivative

at a Local Maximum
Solution

There is a local maximum at x = 0.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.5 A Function with an Undefined Derivative

at a Local Minimum

Locate any local extrema for f (x) = |x| and describe the
behavior of the derivative at the local extremum.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.5 A Function with an Undefined Derivative

at a Local Minimum
Solution
There is a local minimum at
x = 0.

The graph has a corner at

x = 0 and hence, f’ (0) is
undefined.

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 3.3 MAXIMUM AND MINIMUM VALUES

DEFINITION 3.3
A number c in the domain of a function f is called a
critical number of f if f’(c) = 0 or f’(c) is undefined.

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 3.3 MAXIMUM AND MINIMUM VALUES

THEOREM 3.2 (Fermat’s Theorem)

Suppose that f (c) is a local extremum (local maximum or
local minimum). Then c must be a critical number of f .

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.6 Finding Local Extrema of a Polynomial

Find the critical numbers and local extrema of

f (x) = 2x3 − 3x2 − 12x + 5.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.6 Finding Local Extrema of a Polynomial

Solution

Critical numbers:

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.6 Finding Local Extrema of a Polynomial

Solution Critical numbers:

The critical numbers x = –1

and x = 2 correspond to local
extrema.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.7 An Extremum at a Point Where the

Derivative Is Undefined

Find the critical numbers and local extrema of

f (x) = (3x + 1)2/3.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.7 An Extremum at a Point Where the

Derivative Is Undefined
Solution

Of course, f’ (x) ≠ 0 for all x, but f (x) is undefined at

x = −1/3 , which is in the domain of f .

Thus, x = −1/3 is the only critical number of f .

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.7 An Extremum at a Point Where the

Derivative Is Undefined
Solution
x = –1/3 corresponds to the
location of a local minimum
(also the absolute minimum).

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.8 A Horizontal Tangent at a Point That Is

Not a Local Extremum

Find the critical numbers and local extrema of f (x) = x3.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.8 A Horizontal Tangent at a Point That Is

Not a Local Extremum
Solution

f has a horizontal tangent line at x = 0, but does not have a

local extremum there.
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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.9 A Vertical Tangent at a Point That Is Not

a Local Extremum
Find the critical numbers and local extrema of f (x) = x1/3.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.9 A Vertical Tangent at a Point That Is Not

a Local Extremum
Solution

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.10 Finding Critical Numbers of a Rational

Function

Note that the domain of f consists of all real numbers

other than x = −2.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.10 Finding Critical Numbers of a Rational

Function
Solution

f’(x) = 0 for x = 0,−4 and f’(x) is undefined for x = −2.

However, −2 is not in the domain of f and consequently,
the only critical numbers are x = 0 and x = −4.
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 3.3 MAXIMUM AND MINIMUM VALUES

THEOREM 3.3

Suppose that f is continuous on the closed interval [a, b].

Then, each absolute extremum of f must occur at an

endpoint (a or b) or at a critical number.

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 3.3 MAXIMUM AND MINIMUM VALUES

REMARK 3.4
Theorem 3.3 gives us a simple procedure for finding the
absolute extrema of a continuous function on a closed,
bounded interval:

1. Find all critical numbers in the interval and compute

function values at these points.
2. Compute function values at the endpoints.
3. The largest function value is the absolute maximum
and the smallest function value is the absolute
minimum.

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.11 Finding Absolute Extrema on a Closed

Interval

Find the absolute extrema of f (x) = 2x3 − 3x2 − 12x + 5 on

the interval [−2, 4].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.11 Finding Absolute Extrema on a Closed

Interval
Solution
The maximum appears to be
at the endpoint x = 4. The
minimum appears to be at a
local minimum near x = 2.
From example 2.6, the critical
numbers are x = −1 and x = 2.
Both of these are in the
interval [−2, 4].

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.11 Finding Absolute Extrema on a Closed

Interval
Solution
Compare the values at the
endpoints:

and the values at the critical

numbers:

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 3.3 MAXIMUM AND MINIMUM VALUES

EXAMPLE 3.11 Finding Absolute Extrema on a Closed

Interval
Solution

Theorem 2.3 says that the

absolute extrema must be
among these four values.
Thus, f (4) = 37 is the absolute maximum and f (2) = −15 is
the absolute minimum.
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