Maximum amp Minimum

Values Calculus 1

Chapter 41 and 43

Ms Medina

Why is this important

bull An important application of differential calculus are

problems dealing with optimization

Examples

bull What is the shape of a can that minimizes manufacturing

cost

bull What is the maximum acceleration of a space

shuttle

bull What is the radius of a contracted windpipe that

expels air most rapidly during a cough

Definition Absolute Maximum amp

Minimum

Absolute Maximum (global maximum)

A function f has an absolute maximum at c if f(1113093c) ge 11130931113093f(1113093x)

1113093for all x in D where D is the domain of f The number f(1113093c) 1113093

is called the maximum value of f on D

Absolute Minimum (global minimum)

A function f has an absolute minimum at c if f(1113093c)1113093 le 1113093 f(1113093x)

1113093 for all x in D and the number f(1113093c)1113093 is called the minimum

value of f on D

The maximum and minimum values of f are called the extreme

values of f

Find the Maximum and

Minimum

Minimum value f(a)

Maximum value f(d)

Absolute Minimum value at f (x) = 0

No Maximum value

No Minimum value

No Maximum value

Definition Local Maximum amp

Minimum

Local Maximum (relative maximum)

A function f has a local maximum at c if f1113134(c)1113134 ge1113134 f(1113134

x)1113134 when x is near c [This means that f(1113134c)1113134 ge1113134 f1113134

(x)1113134 for all x in some open interval containing c]

Local Minimum (relative minimum)

Similarly f has a local minimum at c if f(1113134c)1113134 le1113134 f(1113134

x)1113134 when x is near c

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Why is this important

bull An important application of differential calculus are

problems dealing with optimization

Examples

bull What is the shape of a can that minimizes manufacturing

cost

bull What is the maximum acceleration of a space

shuttle

bull What is the radius of a contracted windpipe that

expels air most rapidly during a cough

Definition Absolute Maximum amp

Minimum

Absolute Maximum (global maximum)

A function f has an absolute maximum at c if f(1113093c) ge 11130931113093f(1113093x)

1113093for all x in D where D is the domain of f The number f(1113093c) 1113093

is called the maximum value of f on D

Absolute Minimum (global minimum)

A function f has an absolute minimum at c if f(1113093c)1113093 le 1113093 f(1113093x)

1113093 for all x in D and the number f(1113093c)1113093 is called the minimum

value of f on D

The maximum and minimum values of f are called the extreme

values of f

Find the Maximum and

Minimum

Minimum value f(a)

Maximum value f(d)

Absolute Minimum value at f (x) = 0

No Maximum value

No Minimum value

No Maximum value

Definition Local Maximum amp

Minimum

Local Maximum (relative maximum)

A function f has a local maximum at c if f1113134(c)1113134 ge1113134 f(1113134

x)1113134 when x is near c [This means that f(1113134c)1113134 ge1113134 f1113134

(x)1113134 for all x in some open interval containing c]

Local Minimum (relative minimum)

Similarly f has a local minimum at c if f(1113134c)1113134 le1113134 f(1113134

x)1113134 when x is near c

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Definition Absolute Maximum amp

Minimum

Absolute Maximum (global maximum)

A function f has an absolute maximum at c if f(1113093c) ge 11130931113093f(1113093x)

1113093for all x in D where D is the domain of f The number f(1113093c) 1113093

is called the maximum value of f on D

Absolute Minimum (global minimum)

A function f has an absolute minimum at c if f(1113093c)1113093 le 1113093 f(1113093x)

1113093 for all x in D and the number f(1113093c)1113093 is called the minimum

value of f on D

The maximum and minimum values of f are called the extreme

values of f

Find the Maximum and

Minimum

Minimum value f(a)

Maximum value f(d)

Absolute Minimum value at f (x) = 0

No Maximum value

No Minimum value

No Maximum value

Definition Local Maximum amp

Minimum

Local Maximum (relative maximum)

A function f has a local maximum at c if f1113134(c)1113134 ge1113134 f(1113134

x)1113134 when x is near c [This means that f(1113134c)1113134 ge1113134 f1113134

(x)1113134 for all x in some open interval containing c]

Local Minimum (relative minimum)

Similarly f has a local minimum at c if f(1113134c)1113134 le1113134 f(1113134

x)1113134 when x is near c

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Find the Maximum and

Minimum

Minimum value f(a)

Maximum value f(d)

Absolute Minimum value at f (x) = 0

No Maximum value

No Minimum value

No Maximum value

Definition Local Maximum amp

Minimum

Local Maximum (relative maximum)

A function f has a local maximum at c if f1113134(c)1113134 ge1113134 f(1113134

x)1113134 when x is near c [This means that f(1113134c)1113134 ge1113134 f1113134

(x)1113134 for all x in some open interval containing c]

Local Minimum (relative minimum)

Similarly f has a local minimum at c if f(1113134c)1113134 le1113134 f(1113134

x)1113134 when x is near c

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Definition Local Maximum amp

Minimum

Local Maximum (relative maximum)

A function f has a local maximum at c if f1113134(c)1113134 ge1113134 f(1113134

x)1113134 when x is near c [This means that f(1113134c)1113134 ge1113134 f1113134

(x)1113134 for all x in some open interval containing c]

Local Minimum (relative minimum)

Similarly f has a local minimum at c if f(1113134c)1113134 le1113134 f(1113134

x)1113134 when x is near c

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Definition Critical numbers

Critical Numbers

bull A critical number of a function f is a number c in the

domain of f such that either f`rsquo(c)1113135 =1113135 0 or frsquo(c)1113135

does not exist

bull If f has a local maximum or minimum at c then c is a

critical number of f

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Closed Interval Method

Finding the absolute maximum and minimum values of a

continuous function f on a closed interval 1113134a b1113134

1 Find the values of f at the critical numbers of f in 1113134a b

1113134

2 Find the values of f at the endpoints of the interval

3 The largest of the values from Steps 1 and 2 is the

absolute maximum value the smallest of these values

is the absolute minimum value

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Increasing amp Decreasing

Test

Increasing

If frsquo(11131341113134x) gt 0 on an interval

then f is increasing on that

interval

Decreasing

If frsquo(1113134x)1113134 lt1113134 0 on an

interval then f is decreasing

on that interval

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

The First Derivative Test

bull Suppose that c is a critical number of

a continuous function f

bull If frsquo1113134 changes from positive to

negative at c then f has a local

maximum at c

bull If frsquo1113134 changes from negative to

positive at c then f has a local

minimum at c

bull If frsquo does not change sign at c then f

has no local maximum or minimum at

c

bull Example

If frsquo is positive on both sides of c or

negative on both sides

Example of the First Derivative Test

Direction Critical

Point

Direction

+ c -

- c +

- c -

+ c +

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Definition Concave up amp Down

Concave Up

bull If the graph of f lies above

all of its tangents on an

interval I then it is called

concave upward on I

Concave Down

bull If the graph of f lies below

all of its tangents on I it is

called concave downward

on I

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Concavity Test

bull If frdquo(x) gt 111313411131340 for all x in I

then the graph of f is

concave upward on I

bull If frdquo(x) lt 111313411131340 for all x in I

then the graph of f is

concave downward on I

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Definition Inflection Points

bull A point P on a curve y = f(1113134

x)1113134 is called an inflection

point if f is continuous

there and the curve

changes from

bull concave upward to

concave down- ward

bull concave downward to

concave upward

at P

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

The Second Derivative Test

bull Suppose frsquorsquo is continuous near c

bull If frsquo(c)1113134 =1113134 0 and frsquorsquo(c)11131341113134 gt 0then f has a local minimum

at c

bull If frsquo(c) = 111313411131340and frdquo(c) lt 0then f has a local maximum at c

Example of Second Derivative Test

Reference

bull Stewart J Calculus Early Transcendental sixth

edition 2008