Aim: How can we classify relative extrema as either relative minimums or relative maximums?

Aim: Increasing/Decreasing f & 1st Derivative Course: Calculus

Do Now:

Aim: How can we classify relative extrema as either relative minimums or relative maximums?

The height of a ball t seconds after it is thrown upward from a height of 32 feet and with an initial velocity of 48 feet per second.

a. Verify that f(1) = f(2)

b. According to Rolle’s Theorem, what must be the velocity at some time in the interval [1, 2]?


Increasing and Decreasing Functions6

5

4

3

2

1

2 4 6 8

Constant

Incr

easin

gDecreasing

x = a x = b

f’(x) < 0 f’(x) = 0 f’(x) > 0

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

1.


1.


1.


1.


1.


1.


1.


1.


1. If f’(x) > 0, for all x in (a, b), then f is increasing on [a, b]

as x moves to the right



5

4

3

2

1

2 4 6 8

Constant

Incr

easin

gDecreasing

x = a x = b

f’(x) < 0 f’(x) = 0 f’(x) > 0


1.


1.


1.


1.


1.


1.


1.


1.


2. If f’(x) < 0, for all x in (a, b), then f is decreasing on [a, b]



5

4

3

2

1

2 4 6 8

Constant

Incr

easin

gDecreasing

x = a x = b

f’(x) < 0 f’(x) = 0 f’(x) > 0


1.


1.


1.


1.


1.


1.


1.


1.


3. If f’(x) = 0, for all x in (a, b), then f is constant on [a, b]


Model Problem

Find the open intervals on which

is increasing or decreasing.

3 23( )2

f x x x

Find critical numbers2'( ) 3 3f x x x = 0

'( ) 3 1 0f x x x

x = 0, 1

2

1

-1

-2

2

Incr

easi

ng

Decreasing

Incr

easi

ng(0, 0)

(1, -.5)no points where function is undefined


Model Problem

Find the open intervals on which is increasing or decreasing.

Interval

Test Value

Sign of f’(x)

Conclusion

2

1

-1

-2

2

Incr

easi

ng

DecreasingIn

crea

sing

(0, 0)

(1, -.5)

Increasing IncreasingDecreasing

f’(-1) = 6 f’(1/2) = -3/4 f’(2) = 6

x = -1 x = ½ x = 2

- < x < 0 (-, 0)

0 < x < 1(0, 1)

1 < x < (1, )


Guidelines

To Find Intervals on Which a Function is Increasing or Decreasing

Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing

1. Located the critical numbers of f in (a, b) and use to determine test intervals.

2. Determine sign of f’(x) at one test value in each interval

3. Determine status on each interval.


The First Derivative TestLet c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.

1. If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f.

( – ) ( + )

f’(x) < 0 f’(x) > 0a c b


The First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, the f(c) can be classified as follows.

2. If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f.

( + ) ( – )

f’(x) > 0 f’(x) < 0a c b


Model Problem 1

Find the relative extrema of the function

in the interval (0, 2)

1( ) sin2

f x x x

1. Find critical values 1'( ) sin 02

df x x x

dx 1'( ) cos 02

f x x

1cos2

x

5,3 3

x

continuous, differentiable,

no where undefined

4

3

2

1

-1

-2

2 4 6

f x = 0.5x-sin x

0 2π


Model Problem 1


in the interval (0, 2)

1( ) sin2

f x x x

2. Create table of intervals

Interval

Test Value x = /4 x = x = 7/4

Sign of f’(x) f’(/4 ) = < 0 f’() > 0 f’(7/4 ) <0

Conclusion decreasing increasing decreasing

1'( ) cos2

f x x

03

x

5

3 3x

5 23

x


Model Problem 1


in the interval (0, 2).

1( ) sin2

f x x x

is relative minimum3

x

5 is relative maximum3

x

3. Conclusion


Model Problem 2

Find the relative extrema of 2 32( ) 4f x x

2 32'( ) 4 0df x x

dx

1 322'( ) 4 2 03

f x x x

1. Find critical values

6

4

2

5

f x = x2-4 23

continuous, differentiable at all but ±2

1 32

4'( ) 03 4

xf x

x

x = 0

2 4 0x x = ±2

undefined at zeros of denom.

Critical values


Model Problem 2


Interv - < x < -2 -2 < x < 0 0 < x < 2 2 < x <

Test Value x = -3 x = -1 x = 1 x = 3

Sign of f’(x) f’(-3 ) < 0 f’(-1) > 0 f’(1 ) < 0 f’(3 ) > 0

Conclusion decre incre decre incre

Find the relative extrema of 2 32( ) 4f x x

1 32

4'( )3 4

xf x

x


Model Problem 2

3. Conclusion

Find the relative extrema of

1 32

4'( )3 4

xf x

x

6

4

2

5

f x = x2-4 23

2 32( ) 4f x x

4

2

-2

-4

-5 5

f x = 4x

3x2-4 13

f has a relative minimum at (-2, 0) & (2, 0)

f has a relative maximum at 3(0, 16)


Do Now:

Aim: How can we classify relative extrema as either relative minimums or relative maximums?

Find the value or values of c that satisfy

for the function on the

interval [3, 9].

( ) ( ) '( )f b f af c

b a

27( )f x x

x

Find the derivative at each critical pointand determine the local extreme values.

5 2 , 12, 1x x

yx x


Model Problem 3

Find the relative extrema of the function 4

2

1( ) xf x

x

1. Find critical values2 2( )'( ) 0d x x

f xdx

= x2 + x-2

4

3 3

2 12'( ) 2x

f x xx x

2

3

2 1 1 10

x x xx

f’(x) = 0 @ ±1; f’(0) is undefined


Model Problem 3


2

1( ) xf x

x

= x2 – x-2


Interv - < x < -1 -1 < x < 0 0 < x < 1 1 < x <

Test Value x = -2 x = -1/2 x = 1/2 x = 2

Sign of f’(x) f’(-2) < 0 f’(-1/2) > 0 f’(1/2) < 0 f’(3) > 2

Conclusion decre incre decre incre

4

3

2 1'( )

xf x

x


Model Problem 3


2

1( ) xf x

x

= x2 – x-2

f has a relative minimum at (-1, 2) & (1, 2)

8

6

4

2

g x = x4+1

x2

6

4

2

-5

h x = 2x4-1

x3

3. Conclusion 4

3

2 1'( )

xf x

x


Model Problem 4

Neglecting air resistance, the path of a projectile that is propelled at an angle is

where y is the height, x is the horizontal distance, g is the acceleration due to gravity,

v0 is the initial velocity, and h is the initial height. Let g = -32 feet per second per

second, v0 = 24 feet per second, and h = 9 feet. What value of will produce the maximum

horizontal distance?

2

22

0

sec tan 02

02

gy x x h

v


Model Problem 4

2

22

0

sec tan 02

gy x x h

v

use values supplied & Quad. Form.

22

232sec tan 9 02 24

x x

2

2sec tan 9 036

x x

2 2

2

2

tan tan secsec / 18

18cos sin sin 1, 0

x

x

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Aim: How can we classify relative extrema as either relative minimums or relative maximums?

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