4.1 Extreme Values for a function

Absolute Extreme Values(a) There is an absolute maximum value at x = c iff

f(c) f(x) for all x in the entire domain.(a) There is an absolute minimum value at x = c iff

f(c) f(x) for all x in the entire domain.

Relative Extreme Values(a) There is a relative maximum value at x = c iff

f(c) f(x) for all x in some open interval containing c.(a) There is a relative minimum value at x = c iff

f(c) f(x) for all x in some open interval containing c.

Maxima and Minima

•Where f (c) =0•Where f (c ) is undefined•At an endpoint of a closed interval

Locations of Extreme values

**Values in the domain of f where f (c) is zero or is undefined are called critical values of the function.***

If f(x) has a maximum or minimum value at x = c it must occur at one of the following locations:

Maxima and Minima on closed interval for continuous function must exist.

Maximum and minimums

A curve with a local maximum value. The slope at c, is zero.

Local Max

4.3 First Derivative test for Increasing and Decreasing functions

If a function is continuous on [a, b] and differentiable on (a,b)

(a) If f > 0 at each point of (a,b) then f increases on [a,b].

(a) If f < 0 at each point of (a,b) then f decreases on [a,b].

Find the critical points and identify intervals on which f is increasing and decreasing.

First derivative test for Local Extrema

At a critical point x = c

1. f has a local minimum if f changes from negative to positive at c.

2. f has a local maximum if f changes from positive to negative at c.

3. There is no extreme value if the sign of f does not change. Could be a horizontal tangent without direction change.

Figure 3.24: The graph of f (x) = x3 is concave down on (–, 0) and concave up on (0, ).Concavity

Second derivative test for Concavity

A graph is concave up on any interval whereThe second derivative is positive.

A graph is concave down on any interval whereThe second derivative is negative.

A point of inflection for a function Occurs where the concavity changes

Second derivative test for extreme values

If there is a critical value at x = c

and

( ) 0f c

( ) 0f c

( ) 0f c

A local max at x = c

conclusion

A local min at x = c

inconclusive

+ ++

- --

Section 1 / Figure 1

Section 4.3 Figures 11Graph of the curve

34 4xxy

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A

The graph of f (x) = x4 – 4x3 + 10.

Finding intervals of concave up and concave down and Inflection points

What derivatives tell us about the shape of a graph

Figure 1.42: The blue graph of f (x) = x + e–x looks like the graph of g(x) = x (black) to the right of the y-axis and like the graph of h(x) = e–x (red) to the left of the y-axis. (Example 1)

4. 4 Limits to Infinity (End behavior)

What happens to the value of the function when the value of x increases without bound?

What happens to the value of the function when the value of x decreases without bound?limx

limx

Basic limits to infinity

1lim 0x x

1lim 0 0, 0n

x nn

x

lim *0 0 0x n

kk n

x

3

5: lim 0xExample

x

1lim 0x x

lim *0 0 0x n

kk n

x

Figure 1.27: The function in Example 3.

2

2 2 2

2

2 2

5 8 35 5

lim lim3 33 2

x x

x x

x x xx

x x

Divide each term by highest power of x in the denominatorand calculate limits

2

2

5 8 3lim

3 2x

x x

x

As x gets larger and larger, the function gets closer and closer to 5/3.

Figure 1.27: The function in Example 3.

2

3 3 3

3

3 3

5 8 30

lim lim 033 2

x x

x x

x x xx

x x

Divide each term by highest power of x in the denominator and calculate limits

2

3

5 8 3lim

3 2x

x x

x

As x gets larger and larger, the function gets closer and closer to 0.

Figure 1.27: The function in Example 3.

4 2

22 2 2

3

2 2

5 8 35

lim lim33 2

x x

x xxx x x

x

x x

Divide each term by highest power of x in the denominatorand calculate limits

4 2

3

5 8 3lim

3 2x

x x

x

As x gets larger and larger, the function decreases without bound.

Figure 1.27: The function in Example 3.2

2

5 8 3lim

3 2x

x x

x

As x gets larger and larger, the function gets closer and closer to 5/3.

When the limit to infinity exists, at y = L we can say that the line y = L is a horizontal asymptote.

Figure 1.27: The function in Example 3.Horizontal Asymptote

Limits that are infinite (y increases without bound)

41

lim4x x

21

lim2x x

31

lim3x x

An infinite limit will exist as x approaches a finite value when direct substituion produces

0

not zero

If an infinite limit occurs at x = c we have a vertical asymptote with the equation x = c.

Figure 1.29: The function in Example 5(a).Slant Asymptote

As x gets larger and larger, the graph gets closer and closer to the line 2 8

7 49y x

As x gets smaller and smaller, the graphgets closer to the same line. You can use long division to rewrite the given function.

There is a vertical asymptote at x = - 4 / 7

Section 1 / Figure 1

Graphs of the polynomial 235 2332)( xxxxxf 6

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

4.6Different scales

Section 1 / Figure 1

Derivatives of the polynomial 235 2332)( xxxxxf 6

© 2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

Graphs of derivatives

Section 1 / Figure 1

2 3

2 4

( 1)( )

( 2) ( 4)

x xf x

x x

Section 1 / Figure 1

Section 1 / Figure 1

Section 4.6 Figures 19, 20The family of functions

221

2

xxy

221

2

xxy

)2/(1)( 2 cxxxf

The family of functions )2/(1)( 2 cxxxf

1c 0c 1c

2c 3c