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3.3 Increasing and Decreasing Functions and the First Derivative Test
Test for Increasing and Decreasing FunctionsLet be a continuous function on [ , ] and differentiable on (a,b).f a b
1. '( ) 0 in ( , ) is increasing on [ , ].f x x a b f a b
2. '( ) 0 in ( , ) is decreasing on [ , ].f x x a b f a b
3. '( ) 0 in ( , ) is constant on [ , ].f x x a b f a b
3.3 Increasing and Decreasing Functions and the First Derivative Test
x a x b
constant
decreasing increasing
' 0f x ' 0f x ' 0f x x
y
f
Guidelines to Finding Intervals on Which
a Function Is Increasing or Decreasing
Let be continuous on ( , ).f a b
1. Locate the critical numbers in ( , ) and determine test intervals.a b
2. Find the sign of '( ) at a test value in each interval.f x
3. Use the Test for Increasing and Decreasing Functions
to decide whether is increasing or decreasing on each interval.f
3.3 Increasing and Decreasing Functions and the First Derivative Test
Without a calculator, find the open intervals on which3 23
( )2
f x x x
is increasing or decreasing.
Critical Numbers will give us
the location of possible extrema.
2'( ) 3 3f x x x 3 ( 1)x x
0,1 are the critical numbers.c
0
3.3 Increasing and Decreasing Functions and the First Derivative Test
0 and 1 arBecause ' e the onl is defined 's, y critical numb ,ersf R
We simply need to create a table to test the three intervals
determined by the two critical numbers.
Interval 0x 0 1x 1 x Test Value 1x 1/ 2x 2x Sign of '( )f x positive negative positive
Conclusion Increasing Decreasing Increasing
2'( ) 3 3f x x x ( ) f x is
3.3 Increasing and Decreasing Functions and the First Derivative Test
Graph
2'( ) 3 3f x x x
3.3 Increasing and Decreasing Functions and the First Derivative Test
decreasing
increasing
increasing
1
1
11,
2
0,0
3 23( )
2f x x x
Test Points
is strictly monotonic on [a,b] if it is increasing or decreasing on the entire interval.f
3.3 Increasing and Decreasing Functions and the First Derivative Test
3( )f x x( )f x { 0,0 1x
2 , 0x x
21 , 1x x
1
The First Derivative TestLet be a critical number of , which is continuous on
containing .
c f I
c
is differentiable on (except possibly at )f I c
( ) can be classified as follows.f c
3.3 Increasing and Decreasing Functions and the First Derivative Test
'( ) changes from negative to positive at
( ) is a relative minim
1
um of
.
.
f x c
f c f
Relative Minimum
'( ) changes from positive to negative at
( ) is a relative maxim
2
um of
.
.
f x c
f c f
Relative Maximum
Neither relative minimum nor relative maximum
Neither relative minimum nor relative maximum
Applying the First Derivative Test
1Find the relative extrema of ( ) sin on (0,2π).
2f x x x
Critical Numbers?
1'( ) cos 0
2f x x
1 5cos ,
2 3 3x x
(since ' is defined 's, these are the only C 's) Vf R
3.3 Increasing and Decreasing Functions and the First Derivative Test
Interval
Test Value
Sign of '( )f x
Conclusion IncreasingDecreasing
0,3
5,
3 3
5,2
3
/ 4x x 7 / 4x
positivenegative negative
Decreasing
By applying the First Derivative Test,
you can conclude that has a relative
minimum at
f
3x
and a relative maximum at3
.5
x
1'( ) cos
2f x x
( ) f x is
3.3 Increasing and Decreasing Functions and the First Derivative Test
3x
5
3x
0,2
3
1
1sin
2y x x
Relative Min
Relative Max
5
3
5,?
3
,?3
3 3
3 6
3
6 2f
5 35 5 3
6
3
3 6 2f
3 3
6
5 3 3
6
22 3
Find the relative extrema of
( ) ( 4) . f x x
2 1/3
1/32
2'( ) ( 4) 2
34
3 4
Critical #'s @ 0, 2
f x x x
x
x
3.3 Increasing and Decreasing Functions and the First Derivative Test
IncreasingDecreasing
0 0
, 2 2,0 0,2.INT
TV'( )f x
Decreasing Increasing
3x 1x 1x 3x 2,
'( 3)f '( 1)f '(1)f 0 '(3)f 0
( 2,0)
Relative Min at (2, (2))f (2,0)
Relative Min at ( 2, ( 2))f
Relative Max at (0, (0))f (0, 2.520)
2/32( ) 4f x x
3.3 Increasing and Decreasing Functions and the First Derivative Test
( ) f x is
1/32
4'( )
3 4
xf x
x
0, 2x ,
3.3 Increasing and Decreasing Functions and the First Derivative Test
2/32 4y x
4
2
Find the relative extrema of
1( ) .
xf x
x
2 2( )f x x x
3
22'( )f x x
x
3
1'( ) 2f x x
x
3.3 Increasing and Decreasing Functions and the First Derivative Test
3
4
21
x
x
2 2
3
2( 1)( 1)x x
x
'( ) 0 at 1f x x '( ) at 0f x DNE x
3
1'( ) 2
xf x x
3.3 Increasing and Decreasing Functions and the First Derivative Test
IncreasingDecreasing
0 0
, 1 1,0 0,1.INT
TV'( )f x
Decreasing Increasing
2x 1/ 2x 1/ 2x 2x 1,
'( 2)f '( 1/ 2)f '(1/ 2)f 0 '(2)f 0
( 1,2)
(0) DNEf
Relative Min at (1, (1))f (1,2)
Relative Min at ( 1, ( 1))f
2 2
3
2( 1)( 1)'
x xf x
x
4
2
1( )
xf x
x
0, 1x ,
The path of a projectile that is propelled at an angle is
y heighthorizontal distancex
g acceleration due to gravity
0 v initial velocity
22
20
sec(tan ) , 0
22
gy x x h
v
3.3 Increasing and Decreasing Functions and the First Derivative Test
32 feet per second per secondg
0 24 feet per secondv
9 feeth What value of will produce
a maximum horizontal distance?
Write in terms of and find the max.x
3.3 Increasing and Decreasing Functions and the First Derivative Test
22
20
sec(tan )
2
gy x x h
v
32
24
9 0
We may need Bell’s THM!!
22
20
sec(tan )
2
gy x x h
v
Write in terms of and find the max.x 3.3 Increasing and Decreasing Functions and the First Derivative Test
9 0
2
2secta 9
360 nx x
32
24
22
2
sectan tan 4 (9)
36
sec2
36
x
2
2
2tan tan sec
sec /18x
3.3 Increasing and Decreasing Functions and the First Derivative Test
22
2
4sec
tan 936
236
tan ( )
secx
Tangent and secant!!! You must be kidding. Let’s write everything in terms of sine and cosine.
2 2
2 2
18 tan 18 tan sec
sec secx
218cos (sin sin 1)x
2 2
2 2
18 tan 18 tan sec
sec secx
22 2
2
sin sin 118cos 18cos
cos cosx
218sin cos 18cos 1 sinx
0x
Now find the which produces a maximum value of .x
3.3 Increasing and Decreasing Functions and the First Derivative Test
Using the first derivative test would be very tedious.
Let's use technology to find / 0.dx d
35.264
Graph
and find where it equals 0.
dx
d
218cos sin sin 1x
218cos sin sin 1x 0 at 35.3 dx
d
HW 3.3 pp. 176-179/ 10,11,14,15,17,19,23-37odd,41,50-52,55
3.3 Increasing and Decreasing Functions and the First Derivative Test