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

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

1
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?

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
3
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
4
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
5
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
6
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
7
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.
8
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
9
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
10
Model Problem 1
Find the relative extrema of the function in
the interval (0, 2?)
continuous, differentiable, no where undefined
1. Find critical values
11
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
12
Model Problem 1
Find the relative extrema of the function in
the interval (0, 2?).
3. Conclusion
13
Model Problem 2
Find the relative extrema of
1. Find critical values
continuous, differentiable at all but 2
x 0
Critical values
14
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
15
Model Problem 2
Find the relative extrema of
3. Conclusion
16
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.
17
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
18
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
19
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)
20
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?
21
Model Problem 4