Download ation Version - Parkway C-2

Determine where a function is increasing or decreasing
When determining if a graph is increasing or decreasing we always
start from left and use only the x values.
Included and excluded do not apply, we always use ( ).
Increase: (-∞, -2.4)
(1.6, ∞)
Decrease: (-2.4, 1.6)
1.
2.
3.
4.
Objectives:
Be able to define various vocabulary terms needed to be successful
in this unit.
Be able to understand the definition of extrema of a function on an
interval and “The Extreme Value Theorem”.
Be able to find the relative extrema and critical numbers of a
function.
Be able to find extrema on a closed interval.
Critical Vocabulary:
Extrema, The Extreme Value Theorem, Critical Numbers
I. Vocabulary
Extrema (Plural of Extreme): This means we are talking about maxima
(plural of maximum) and minima (plural
of minimum) of a function.
Interval: Means we are talking about a part of a function,
denoted by interval notation (a, b) or [a, b]
Open Interval Closed Interval
II. Extrema of a Function
Let f be defined on an interval I containing c
1. f(c) is the MINIMUM of f on I if f(c) ≤ f(x) for all x in I
2. f(c) is the MAXIMUM of f on I if f(c) ≥ f(x) for all x in I
The minimum and maximum of a function on an interval are
the extreme values, or extrema, of the function on the
interval. The minimum and maximum of a function on an
interval are also called the absolute minimum and absolute
maximum on the interval.
Example 1: Function: f(x) = x2 + 3, Interval: [-2, 2], Let c = 0
f(c): f(0) = 3
f(-2) = 7 f(-1) = 4 f(1) = 4 f(2) = 7
f(c) is less than or equal to all values of f(x) on the interval
making f(c) the MINIMUM of f
II. Extrema of a Function
If f is continuous on a closed interval [a, b], then f has both
a minimum and maximum on the interval.
To illustrate this, we will look at the graph of f(x) = x2 + 1
on the following intervals:
[-1, 2]
Min: (0, 1)
Max: (2, 5)
(-1, 2)
Min: (0, 1)
No Max
[-1, 2]
No Min
Max: (2, 5)
III. Relative Extrema and Critical Numbers
Point where a graph changes its behavior (increasing/decreasing)
help in determining the maximum and minimum values of a graph.
Relative Minimum: The smallest point of the graph in a given area.
Relative Maximum: The largest point of the graph in a given area.
Relative
Max “Hill”
Relative
Max “Hill”
Relative
Min
“Valley”
Relative
Min
“Valley”
III. Relative Extrema and Critical Numbers
1. If there is an open interval containing c on which f(c) is
a maximum, then f (c) is called a relative maximum of f.
2. If there is an open interval containing c on which f(c) is
a minimum, then f (c) is called a relative minimum of f.
The plural of relative maximum is relative maxima and the
plural of relative minimum is relative minima.
III. Relative Extrema and Critical Numbers
Example 2: Find the value of the derivative at each of the relative
extrema shown in the graph

 
f ( x)  9 x 2  27 x 3
f ( x)  9 x 1  27 x 3
f ' ( x)  9 x 2  81x 4
f ' ( x) 
 9 81
 4
2
x
x

 9 x 2  81
f ' ( x) 
x4

9  x2  9
f ' ( x) 
x4


9  (3) 2  9
f ' (3) 
(3) 4
f ' (3)  0


9 x2  3
f ( x) 
x3
When the derivative is zero, we call the x-value
associated with it a CRITICAL NUMBER.
III. Relative Extrema and Critical Numbers
Example 3: Find any critical numbers algebraically: f(x) = x2(x2 - 4)
1st: Find the derivative

f ( x)  x 2 x 2  4

f ( x)  x 4  4 x 2
f ' ( x)  4 x 3  8 x
2nd: Set f’(x) = 0
0  4 x3  8x

0  4x x2  2
4x  0
x0

x2  2  0
x2  2
x 2
3rd: Check for any places where the derivative is undefined
IV. Finding Extrema on a Closed Interval
To find the extrema of a continuous function f on a closed
interval [a, b], use the following steps:
1. Find the critical numbers of f in (a, b)
2. Evaluate f at each critical number in (a, b)
3. Evaluate f at each end point in [a, b]
4. The least of these values is the minimum and
the greatest is the maximum.
IV. Finding Extrema on a Closed Interval
Example 4: Locate the absolute extrema of the function on the
x2
closed interval
f ( x) 
x2  3
1st: Find the critical numbers (0,0)
f ( x)  x 2 x 2  3
g ( x)  x 2
, [1,1]
Minimum
1
g ' ( x)  2 x
h( x)  x 2  3
1
h( x)  x 2  3  2 x
2
f ' ( x)  2 xx  3  2 x x  3
1
2
3

f ' ( x)  2 xx 2  3 x 2  3  x 2
2
f ' ( x) 
2nd
1
2

6x
6x  0
x  3 x  0
: Evaluate at the endpoints
1

Left End point:   1, 
4

2
2
 1
 4
Right End point: 1, 
Maximum
IV. Finding Extrema on a Closed Interval
Example 5: Locate the absolute extrema of the function on the
closed interval
f ( x)  x1/ 3 , [1,1]
1st: Find the critical numbers (0,0)
f ( x) 
f ( x) 
1 2 3
x
3
1
33 x 2
Critical number: x = 0
2nd : Evaluate at the endpoints
Left End point:  1,1
Minimum
Right End point: 1,1
Maximum
Page 319-321 #7-27 odd, 33, 43, 45, 47
Practice Assessment
1. Find all the relative extrema of the function f(x) = x4 – 8x2
2 3 3 2
x  x  9 x  2, [3,5]
3
2
x
f ( x)  2
, [3,3]
x 1
2. Find the absolute extrema of the function: f ( x) 
3. Find the absolute extrema of the function:
Practice Assessment
1. Find all the relative extrema of the function f(x) = x4 – 8x2
f ' ( x)  4 x 3  16 x
x  0 : (0,0)
x  2 : (2,16)
Relative Maximum
Relative Minimum
x  2 :  2,16 Relative Minimum
Practice Assessment
2
3
2. Find the absolute extrema of the function: f ( x)  x 3  x 2  9 x  2, [3,5]
3
f ' ( x)  2 x 2  3 x  9
x  3 : 3,  41 2
Minimum
x   3 2 : ( 3 2 , 79 8) Maximum
x  3 :  3, 5 2
x  5 : 5,17 6
2
Practice Assessment
3. Find the absolute extrema of the function:
f ' ( x) 
 x2 1
x
2
x  1 : 1,1 2
x  1: (1,1 2)
x  3 :  3, 3 10
x  3 : 3, 3 10

1
2
Maximum
Minimum
f ( x) 
x
, [3,3]
2
x 1