Warm Up
Finish Chapter 4 Test
5.1 – Extreme Values
• Goal: I will find Absolute and Local extreme of functions.
Definition
Absolute extreme values are either maximum or
minimum points on a curve.
They are sometimes called global extremes.
They are also sometimes called absolute extrema.
(Extrema is the plural of the Latin extremum.)
Formal Definition
Definition Absolute Extreme Values
Let f be a function with domain D. Then f (c) is the
a. absolute maximum value on D if and only
if f(x) < f (c) for all x in D.
b. absolute minimum value on D if and only
if f(x) > f (c) for all x in D.
Example
Extreme values can be in the interior or the
end points of a function.
4
No Absolute
Maximum
3
yx
2
2
D   ,  
Absolute
Minimum of 0 at x=0
1
-2
-1
0
1
2
Example
yx
2
Absolute
Maximum of 4 at x=2
4
D  0, 2
3
2
1
-2
-1
0
1
2
Absolute Minimum of 0 at x=0
Example
4
yx
2
Absolute
Maximum of 4 at x=2
3
D   0, 2
2
1
-2
-1
0
1
No Minimum
2
Example
4
No
Maximum
3
2
1
yx
2
D   0, 2 
-2
-1
0
1
No Minimum
2
Extreme Value Theorem:
If f is continuous over a closed interval [a,b],
then f has both a maximum and minimum
value over that interval.
Maximum & minimum
at interior points
Maximum &
minimum
at endpoints
Maximum at
interior point,
minimum at
endpoint
Definition
Local Extreme Values (also called relative extrema):
A local extreme is a point that is a
max or min of its “neighborhood” or
of any points nearby
Example
Absolute maximum
(also local maximum)
Local maximum
Local minimum
Local minimum
Absolute minimum
(also local minimum)
Example
Absolute maximum
(also local maximum)
Local maximum
Local minimum
Notice that local extremes in the interior of the function occur where 𝑓′ is
zero or 𝑓′ is undefined.
Definition
A point in the interior of the domain of f at which
𝑓 ′ = 0 or 𝑓′ does not exist is a critical point.
Extreme values only occur at critical points or end
points.
How to find Extrema
1. Find the derivative.
2. Find where it equals 0 or does not exist.
3. Check those values and endpoints by plugging them into the
original equation. Check to see what values are the max and min.
Example
2
3
Find the absolute max and min values of 𝑓 𝑥 = 𝑥 on the interval [-2,3]
2
f  x  x
3

1
3
There are no values of x that will make
the first derivative equal to zero.
The first derivative is undefined at x=0
f  x 
2
33 x
Because the function is defined over a
closed interval, we also must check the
endpoints.
Example continued
• Let’s plug in the critical point and endpoints into f(x).
•𝑓 0 =0
• 𝑓 −2 = −2
2
3
= 1.587
2
3
• 𝑓 3 = 3 = 2.08
• Looking at our list, we can see the absolute min is 0 at x=0 and the
absolute max is 2.08 at x=3.
Answer
• Abs. Min: (0,0)
• Abs. Max: (3, 2.08)
You Try
Find the absolute maximum and minimum of the function
f ( x)  2 x3  5x 2  4 x  2, on [1,2]
f ' ( x)  6 x  10 x  4
2
Find the critical numbers.Hint: Use the quadratic formula.
Answer
f ( x)  2 x3  5x 2  4 x  2, on [1,2]
Check endpoints and critical numbers
f x 
x
 1  13
2  26
3 27
1 1
The absolute maximum is 2 when x = -2
The absolute minimum is -13 when x = -1
2
2
Try again! 
Find the absolute maximum and minimum of the function
x 3
f ( x) 
, on [0,3]
x 1
2
Find the critical numbers
0  x2  2 x  3
0  ( x  3)( x  1)
( x  1)( 2 x)  ( x 2  3)(1)
f ' ( x) 
( x  1)2
x2  2 x  3
f ' ( x) 
( x  1)2
x  3 or x 1
Answer
Find the absolute maximum and minimum of the function
x 3
f ( x) 
, on [0,3]
x 1
2
Check endpoints and critical numbers
The absolute maximum is 3 when x = 0, 3
The absolute minimum is 2 when x = 1
x
f x 
0
3
1
2
3
3
Example
Find the absolute maximum and minimum of the function
f ( x)  sin x  sin 2 x , on 0,2 
f ' ( x)  cos x  2 sin x cos x
Find the critical numbers
0  cos x  2 sin x cos x
0  cos x(1  2 sin x)
cos x  0
x
 3
,
2 2
1  2 sin x  0
x
 5
,
6 6
Solution
Find the absolute maximum and
minimum of the function
f ( x)  sin x  sin 2 x , on 0,2 
The absolute maximum is 1/4 when x = /6, 5/6
The absolute minimum is –2 when x =3/2
x
f x 
0
0

1
4
6

2
5
6
3
2
2
0
1
4
2
0
Remember
Critical points are not always extremes!
2
yx
3
1
-2
-1
0
-1
-2
1
2
f0
(not an extreme)
Homework
• 5.1 on Calendar