EXTREMA ON AN EXTREMA ON AN INTERVALINTERVAL

EXTREMA ON AN EXTREMA ON AN INTERVALINTERVAL

Section 3.1Section 3.1

When you are done with your homework, you should

be able to…• Understand the definition of

extrema of a function on an interval

• Understand the definition of relative extrema of a function on an open interval

• Find extrema on a closed interval

EXTREMA OF A FUNCTION

1c

1f c

2f c

2ca b

DEFINITION OF EXTREMA

Let f be defined on an open interval I containing c.

1. is the minimum of f on I if for all x in I.

2. is the maximum of f on I if for all x in I.

f c f c f c

f c

f c f x

EXTREMA CONTINUED…

• The minimum and maximum of a function on an interval are the extreme values, or extrema of the function on the interval

• The singular form of extrema is extremum

• The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval

EXTREMA CONTINUED…

• A function does not need to have a maximum or minimum (see graph)

• Extrema that occur at endpoints of an interval are called endpoint extrema

3

2

1

-2 2

f x = x2

THE EXTREME VALUE THEOREM

If f is continuous on a closed interval

then f has both a minimum and a maximum on the interval.

, ,a b

DEFINITION OF RELATIVE EXTREMA

1. If there is an open interval containing c on which is a maximum, then is called a relative maximum of f, or you can say that f has a relative maximum at

2. If there is an open interval containing c on which f is a minimum, then is called a relative minimum of f, or you can say that f has a relative minimum at

, .c f c

f c

, .c f c

f c

f c

Find the value of the derivative (if it exists) at the indicated extremum.

0.00.0

cos ; 2, 12

xf x

Find the value of the derivative (if it exists) at the indicated extremum.

0.00.0

2 2 33 1; ,

3 3r s s s

Find the value of the derivative (if it exists) at the indicated extremum.

4 ; 0,4f x x

0 0

0 0

lim

4 40lim lim 1

0

4 40lim lim 1

0Therefore, 0 does not exist

x c

x x

x x

f x f cf x

x c

xf x f

x x

xf x f

x xf

DEFINITION OF A CRITICAL NUMBER

Let f be defined at c.

1. If , then c is a critical number of f.

2. If f is not differentiable at c, then c is a critical number of f.

' 0f c

Locate the critical numbers of the

function.

A.

B.

C.

D. None of these

2, 0c c

2 4x

f xx

2c

2, 0c c

Locate the critical numbers of the function.

A.

B.

C.

D. None of these

2sec tan , 0, 2f 3 7 11

, , , 2 2 6 6

c c c c

7 11,

6 6c c

4 5,

3 3c c

THEOREM: RELATIVE EXTREMA OCCUR ONLY AT CRITICAL

NUMBERS

If f has a relative maximum or minimum at

, then c is a critical number of

f.

x c

GUIDELINES FOR FINDING EXTREMA ON A CLOSED

INTERVALTo find the extrema of a continuous function

f on a closed interval , use the following steps.

1. Find the critical numbers of in .2. Evaluate f at each critical number in

.3. Evaluate f at each endpoint of .4. The least of these outputs is the

minimum. The greatest is the maximum.

,a b

,a b

,a b

The maximum of a function that is

continuous on a closed interval at two different values in the interval.

A. TrueB. False