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Aim How can we classify relative extrema as

either relative minimums or relative maximums?

Do Now

- 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. - Verify that f(1) f(2)
- According to Rolles Theorem, what must be the

velocity at some time in the interval 1, 2?

Increasing and Decreasing Functions

x a

x b

as x moves to the right

Increasing

Decreasing

Constant

f(x) 0

f(x) gt 0

f(x) lt 0

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1. If f(x) gt 0, for all x

in (a, b), then f is increasing on a, b

Increasing and Decreasing Functions

x a

x b

Increasing

Decreasing

Constant

f(x) 0

f(x) gt 0

f(x) lt 0

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 2. If f(x) lt 0, for all x

in (a, b), then f is decreasing on a, b

Increasing and Decreasing Functions

x a

x b

Increasing

Decreasing

Constant

f(x) 0

f(x) gt 0

f(x) lt 0

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 1.

Let f be a function that is continuous on the

closed interval a, b and differentiable on the

open interval (a, b). 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.

Find critical numbers

0

x 0, 1

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

-? lt x lt 0 (-?, 0)

0 lt x lt 1 (0, 1)

1 lt x lt ? (1, ?)

x -1

x ½

x 2

f(-1) 6

f(1/2) -3/4

f(2) 6

Increasing

Increasing

Decreasing

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

- 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 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, 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) lt 0

f(x) gt 0

a

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) gt 0

f(x) lt 0

a

c

b

Model Problem 1

Find the relative extrema of the function in

the interval (0, 2?)

continuous, differentiable, no where undefined

1. Find critical values

Model Problem 1

Find the relative extrema of the function in

the interval (0, 2?)

2. Create table of intervals

Interval

Test Value x ?/4 x ? x 7?/4

Sign of f(x) f(?/4 ) lt 0 f(?) gt 0 f(7?/4 ) lt0

Conclusion decreasing increasing decreasing

Model Problem 1

Find the relative extrema of the function in

the interval (0, 2?).

3. Conclusion

Model Problem 2

Find the relative extrema of

1. Find critical values

continuous, differentiable at all but 2

x 0

Critical values

Model Problem 2

Find the relative extrema of

2. Create table of intervals

Interv -? lt x lt -2 -2 lt x lt 0 0 lt x lt 2 2 lt x lt ?

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

Sign of f(x) f(-3 ) lt 0 f(-1) gt 0 f(1 ) lt 0 f(3 ) gt 0

Conclusion decre incre decre incre

Model Problem 2

Find the relative extrema of

3. Conclusion

Aim How can we classify relative extrema as

either relative minimums or relative maximums?

Do Now

Find the value or values of c that satisfy for

the function on

the interval 3, 9.

Model Problem 3

Find the relative extrema of the function

x2 x-2

1. Find critical values

f(x) 0 _at_ 1 f(0) is undefined

Model Problem 3

Find the relative extrema of the function

x2 x-2

2. Create table of intervals

Interv -? lt x lt -1 -1 lt x lt 0 0 lt x lt 1 1 lt x lt ?

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

Sign of f(x) f(-2) lt 0 f(-1/2) gt 0 f(1/2) lt 0 f(3) gt 2

Conclusion decre incre decre incre

Model Problem 3

Find the relative extrema of the function

x2 x-2

3. Conclusion

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

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?

Model Problem 4

use values supplied Quad. Form.