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Class

Meeting#|7

Pointq of Infle,etion

-In

the

business

world,

important

trends &

ehanges occur at

these

points.

-Margins

change

direction at tlese

points.

*Consider

the

following

gaph:

-Look

at

the

slopes

of the tangent

lines

as x

increases. Note that

these

slopes

decrease in the region where the

graph

is

concave down and

increase in

the regionwhere

the

graph

is

concave

up.

Point P

is

where

the slopes stop

decreasing and

start

to

increase. The

concavity changes

there. This

is

an

inflection

point.

a

$mooth

curye,

places

where f

"(x)=g

are

nossible

inflection

points.

Recall

y-14

61

r0

(review

again)

-Ifthere

is

a

concavity

change

at a

point

and

the curve

is

smooth, then

f

"(x)=0.

gSurooth'means

thx a

derivative exists

atthat

point

(i.e.,

curve is differentiable

tlere)

How to locatq

ppints.,qf

f-nfleFtion

*Given

{x),

find

f

"(x).

Find

where

f

"1x1:6

and

justify

a

Brgn

change at these

points.

To

get

the

points,

substitute

the

x-values found

when

solving

f

"(x)=0

into

the equatioa for

f(x),

to

pick

up

the

y-values

of the

points.

Example:

Find

x-values for which the

curve,

f{x}

:

*'

-

g*'

-

48x

+

52, is

concave up and concave down.

Find

atl

points

of

inflection

and

sketch

the curve.

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