4.3Extreme Values of Functions

Greg Kelly, Hanford High School, Richland, Washington

Borax Mine, Boron, CAPhoto by Vickie Kelly, 2004

Local Extreme Values:

A local maximum is the maximum value within some open interval.

A local minimum is the minimum value within some open interval.

Local maximum

Local minimum

Absolute maximum(also local maximum)

Local extremes are also called relative extremes.

Local maximum

Local minimum

Notice that local extremes in the interior of the function occur where is zero or is undefined.f f

Absolute maximum(also local maximum)

Local Extreme Values:

If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, then

0f c

f

Critical Point:A point in the domain of a function f at whichor does not exist is a critical point of f .

Critical points where are called stationary points.

0f f

Note:Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

0f

Critical points are not always extremes!

3y x

0f (not an extreme)

1/3y x

is undefined.f (not an extreme)

The First Derivative Test

Let c be a critical point of a function f that is continuous on some open interval containing c.If f is differentiable on the interval (except possibly at c), then

1. If changes from negative to positive at c, then f(c) is a relative minimum.

( )f x

2. If changes from positive to negative at c, then f(c) is a relative maximum.

( )f x

Example:Use the first derivative test to find the relative extrema of: 3 23 4y x x

23 6y x x

0ySet

20 3 6x x

20 2x x

0 2x x

0, 2x

First derivative test:

y0 2

0 0

21 3 1 6 1 3y negative

21 3 1 6 1 9y positive

23 3 3 6 3 9y positive

Possible extreme at .0, 2x

We can use a chart to organize our thoughts.

Example:Graph 23 23 4 1 2y x x x x

23 6y x x

0ySet

20 3 6x x

20 2x x

0 2x x

0, 2x

First derivative test:

y0 2

0 0

maximum at 0x

minimum at 2x

Possible extreme at .0, 2x

Use the first derivative test to find the relative extrema of: 3 23 4y x x

23 6y x x First derivative test:

y0 2

0 0

NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation!

There is a local maximum at (0,4) because for all x in and for all x in (0,2) .

0y( ,0) 0y

There is a local minimum at (2,0) because for all x in(0,2) and for all x in .

0y(2, )0y

The Second Derivative Test (the easier way!!)

If x = c is a critical point such that , and the second derivative exists on the interval containing c, then

1. If then f(c) is a relative minimum.( ) 0f c

2. If then f(c) is a relative maximum.( ) 0f c

If , the test fails. In such cases you have to use the First Derivative Test.

( ) 0f c

( ) 0f c

Because the second derivative atx = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum.

Example:Graph 23 23 4 1 2y x x x x

23 6y x x Possible extreme at .0, 2x

6 6y x

0 6 0 6 6y

2 6 2 6 6y Because the second derivative atx = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

Use the second derivative test to find the relative extrema of: 3 23 4y x x