Class

Meeting #18

. More

on Maxima

&

Minima

-fmding

ihe absolute

ma*imum

or minimum of a

functiotr over an

interval.

-This

is

also oalled

the

elobal

ma:dmum

or

minimum.

-Iocate

the

critical

values.

Look

here:

1) where f

'(x;:9.

2) where

f

'(x)

does

not

exist

(cusps)..Functions

with cusps have more

complicated

equation forms.

3)

end

points

of

the

irtervals

lbr

the

ihdependent

variable

([zually

x). The

ftnctions

values here

are often the

highest or lowest

points

6n

the

graph.

-MgltiEqpgrtatrtkyou

need

to

compsre the values

ofthe function at these

places

to

determine

which

point

yields

the

largest

or

smallest value.

i

Efampte:

Consider a

function

defined over

[a;b]

with

the fonowing

d<etch.

!

,

Criticat

points

to

test occur at

lrq

f,

d,

g

and

b.

-The

x-values

q

d,

&

c

are

found by

setting

f

'(xFO

and solving

for

the

x's. The

end

points

at

rFa

and

x=b

are

given.

-At:r=f

and:rc,

the

2d

derivative

will be

positive

since the

curve is

concave

up

there.

Sq these

x-vafues

produce

local

minimum

points.

To

get

the

points,

zubstitute

the x-values

into

the

equatiou for

the

fuirction

These

are the

y-values

ofthe

points.

Remember,

a

point

oonsists

of

an ordered

pair

of numbers.

,.

-find

f(a)

and

(b)

and

comparc

their

values

to'the

other

firnctiooal values (y-values)

found

,1

-Iu

this

sketch

the

absolut€ ma:dmum

is

at x=q

the

absolute

minirmrm

is

at :rc,

a

local

fielative)

mudmum

occurs

at :rd and

a local

minimum

at tFc

Note that

an absolute

extfeme

is

automatically

a

local

extreme.

Calculus

teachers disagrrc

on

whether

or

not

the

end

points

should

be

called

local

urtreme, I

thitrk

they

should.