Section 4.

3 Derivatives And Graphing

flex _O
of
0
f x
f 17 0

P
f4 7 0

where f Ex o
fix is increasing
where fix so
f x is decreasing

the second derivative

what About
f Cx so

fix 0

I
0
f ex o
 where f Cx o
f x is concave
up
down where f ex o
f ex is concave

fcx changes from

where
The point is
to concave down
concave up
the point of inflection
call

v A point
inflection
f DNE
f x o of x

3
3 2
72 5
Ex fix 2

A where is fox increasing decreasing

f x 6 2 6 72

0 6 XZ X 12

O 6 X 4 x 3
Critical Values 11 4 and X 3

0
y y 0

I
3 4

Pick a number in each of the

test each number

and
three regions
if f an is positive or negative
to see

48
TestX fC 4 6C412 6C 41 72

72 72
f'co 6672 6 o
Test
48
f best 615 72
Tests s
 t t t t t t t t t t t t t t

fan increasing C 00 3 U 4,007

fex C 3,4
decreasing

s
A

I f
d

with
The graph agrees with
the math told us About fix
what
 fix concave down
B where is up

flex 6
2
6 72

f x 12 6

12 6
0
l2x 6

X I
2

o
y
1

interval to see if f ex
Test each
is positive
or Negative

O t I
2

Test f co 1267 6 6
 Test f o 124 6 6

t t t t t

fix concave down 00

f x concave up
00

MAX

v min

down where
Notice that fan is concave

maximum And faa is convave

fix has a

minimum
where f x has a
up
 called the Second Derivative Test
This is

Second Derivative Test to

Use the

and maximums for

find minimums
o T
2x on
X t sin
y
cos 2x 2
It
y
It 2 cos 2x
O
213431

Cos 2x I 4
2x cos't'z
2x IT
2x 3
one
3

IT
Critical Values X ZI ore
3

to find out if there

use
Now y
At those
or minimums
Are maximums
two critical values
 SMC2 7.4
y
3.464 Concave Down
Ig 4 sin
y

II 4 sin 3.464 Concave Up

y
mat
This means that
the graph should
Npm in

look something I
y
this 3 3
like

to verify
Now graph fan

21173
Tfg
I 1

min 2.094 1.228

MAX 1.047 1.913
 continuous function fax
EI Draw a

with All of the following properties

f 2 3
fC 3 2 fc 1 I f 1 2

f C37 DNE f C17 0 f 127 0

f x 0 only on C 1,1

3
2
I

s.it I

u r
u v

the first derivative

Now work with

Put if f 17 0

And use ord if f DNE

 v
r
u

work with f X
Now

means that fix

i
between X l and x
is concaveu up only
it is concaven down
otherwise
on C 1,1 the rest is
concaveup concave down

in
t