Review 3.1-3.3

- Increasing or Decreasing- Relative Extrema- Absolute Extrema

- Concavity

-3 -2 -1 1 2 3 4 5 6

-1

1

2

3

4

5

6

7

8

x

y

Increasing/Decreasing/Constant

-3 -2 -1 1 2 3 4 5 6

-1

1

2

3

4

5

6

7

8

x

y

Increasing/Decreasing/Constant

.,on increasing is then

,, intervalan in of each valuefor 0 Ifbaf

baxxf

.,on decreasing is then

,, intervalan in of each valuefor 0 Ifbaf

baxxf

.,on constant is then

,, intervalan in of each valuefor 0 Ifbaf

baxxf

Increasing/Decreasing/Constant

Generic Example

-3 -2 -1 1 2 3 4 5 6

-1

1

2

3

4

5

6

7

8

x

y

( ) 0f x

( )f x DNE

The corresponding values of x are called Critical Points of f

Critical Points of f

a. ( ) 0f c

A critical number of a function f is a number c in the domain of f such that

b. ( ) does not existf c

ExampleFind all the critical numbers of

When set = 0Excluded values

Determine where the function is increasing or decreasing:

-1 0

Plug into the derivate around each critical #

- - - - - + + + + + +

Increasing: (-1, ∞)Decreasing: (-∞, -1)

ExampleFind all the critical numbers of

When set = 0Excluded values

Determine where the function is increasing or decreasing:

-1 0 1

Plug into the derivate around each critical #

Not in Domain + + + - - - - - Not in Domain

Increasing: (-1, 0)Decreasing: (0, 1)

Example

3 3( ) 3 .f x x x

2

233

1( )3

xf xx x

Find all the critical numbers of

0, 3x

When set = 0 1x Excluded values

Determine where the function is increasing or decreasing:

-1 0 1

Plug into the derivate around each critical #

+ + + + + + - - - - - - - - - - + + + + + +

Increasing: (- ∞, -1) U (1, ∞)Decreasing: (-1, 1)

Graph of 3 3( ) 3 . f x x x

-2 -1 1 2 3

-3

-2

-1

1

2

x

y

Local max. 3( 1) 2f

Local min. 3(1) 2f

If the price of a certain item is p(x) and the total cost to produce x units is C(x), at what production levels is profit increasing and decreasing?

Now find P’(x)

Now test around 18, -2

Increasing: (0, 18)Decreasing: (18, ∞)

Relative ExtremaA function f has a relative (local) maximum at x = c if there exists an open interval (r, s) containing c such that f (x) = f (c)

Relative Maxima

Relative ExtremaA function f has a relative (local) minimum at x = c if there exists an open interval (r, s) containing c such that f (c) = f (x)

Relative Minima

The First Derivative Test

left right

f (c) is a relative maximum

f (c) is a relative minimum

No change No relative extremum

Determine the sign of the derivative of f to the left and right of the critical point.

conclusion

The First Derivative Test.16)( 23 xxxf

2( ) 3 12 0f x x x

Find all the relative extrema of

0)4(3 xx4,0x

0 4

+ 0 - 0 +

Relative max. f (0) = 1

Relative min. f (4) = -31

f

f

Excluded Values: None

-2 -1 1 2 3 4 5 6 7 8 9 10

-35

-30

-25

-20

-15

-10

-5

5

x

y

(4,f(4)=-31)

(0,f(0)=1)

The First Derivative Test

The First Derivative TestFind all the relative extrema of

Excluded Values: None

-1 0 1

+ + + - - - - - - - - - - + + +

Rel. Min. (1, -2)Rel. Max. (-1, 2)

-1 0 1

+ ND + 0 - ND - 0 + ND +

Relative max. Relative min.

f

f

0, 3x

1x

Exclude Values:3(1) 2f

3

3( 1) 2f

3

3 3( ) 3 .f x x x

Example from before:

2

233

1( )3

xf xx x

-2 -1 1 2 3

-3

-2

-1

1

2

x

y

Rel. max. 3( 1) 2f

Rel. min. 3(1) 2f

Graph of 3 3( ) 3 . f x x x

Absolute Extrema

Absolute Minimum

Let f be a function defined on a domain D

Absolute Maximum

The number f (c) is called the absolute maximum value of f in D

A function f has an absolute (global) maximum at x = c if f (x) = f (c) for all x in the domain D of f.

Absolute Maximum

Absolute Extrema

c

( )f c

Absolute Minimum

Absolute ExtremaA function f has an absolute (global) minimum at x = c if f (c) = f (x) for all x in the domain D of f.

The number f (c) is called the absolute minimum value of f in D

c

( )f c

Finding absolute extrema on [a , b]

1. Find all critical numbers for f (x) in (a , b).2. Evaluate f (x) for all critical numbers in (a , b).3. Evaluate f (x) for the endpoints a and b of the

interval [a , b]. 4. The largest value found in steps 2 and 3 is the

absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a , b].

ExampleFind the absolute extrema of 3 2 1( ) 3 on ,3 .

2f x x x 2( ) 3 6 3 ( 2)f x x x x x

Critical values of f inside the interval (-1/2,3) are x = 0, 2

(0) 0(2) 4

1 72 8

3 0

ff

f

f

Absolute Max.

Absolute Min.Evaluate

Absolute Max.

ExampleFind the absolute extrema of 3 2 1( ) 3 on ,3 .

2f x x x

Critical values of f inside the interval (-1/2,3) are x = 0, 2

Absolute Min.

Absolute Max.

-2 -1 1 2 3 4 5 6

-5

ExampleFind the absolute extrema of 3 2 1( ) 3 on ,1 .

2f x x x 2( ) 3 6 3 ( 2)f x x x x x

Critical values of f inside the interval (-1/2,1) is x = 0 only

(0) 01 72 8

1 2

f

f

f

Absolute Min.

Absolute Max.

Evaluate

-2 -1 1 2 3 4 5 6

-5

ExampleFind the absolute extrema of 3 2 1( ) 3 on ,1 .

2f x x x 2( ) 3 6 3 ( 2)f x x x x x

Critical values of f inside the interval (-1/2,1) is x = 0 only

Absolute Min.

Absolute Max.

ConcavityLet f be a differentiable function on (a, b).

1. f is concave upward on (a, b) if f ' is increasing on aa(a, b). That is f ''(x) > 0 for each value of x in (a, b).

concave upward concave downward

2. f is concave downward on (a, b) if f ' is decreasing on (a, b). That is f ''(x) < 0 for each value of x in (a, b).

Inflection PointA point on the graph of f at which f is continuous and concavity changes is called an inflection point.

To search for inflection points, find any point, c in the domain where f ''(x) = 0 or f ''(x) is undefined.

If f '' changes sign from the left to the right of c, then (c, f (c)) is an inflection point of f.

Example: Inflection Points

.16)( 23 xxxf2( ) 3 12f x x x

Find all inflection points of

( ) 6 12f x x Possible inflection points are solutions of a) ( ) 0 b) ( ) 6 12 0 no solutions 2

f x f x DNEx

x

2

- 0 +

Inflection point at x 2

f

f

-2 -1 1 2 3 4 5 6 7 8 9 10

-35

-30

-25

-20

-15

-10

-5

5

x

y

(4,f(4)=-31)

(0,f(0)=1)