MAXIMUM AND MINIMUM
VALUES
• A critical number is the value of x that makes
f ′(x) = 0 or f ′(x) undefined
• The absolute minimum of a function is the very
lowest point on the graph over a given interval of
x.
• The absolute maximum of a function is the very
highest point on the graph over a given interval
of x.
• Local minimums and maximums are turning
points on the graph. They are at the values of x
for which f ′(x) = 0.
1
This function is graphed over the interval -2 < x < 7
The absolute maximum value is 5
The local maximum values are 2 and 3
The local minimum values are -2 and 1
The absolute minimum value is -2
2
Procedure for Finding the Absolute Maximum and
Minimum Values of a Continuous Function on a
Closed Interval [a, b]
1. Find the values of f (x) where f ' (x) = 0
2. Find the values of f (a) and f (b).
3. The largest of the values from above is the absolute
maximum value
4. The smallest of these values if the absolute
minimum value
3
EXAMPLE 1
Graph the function f(x) = x2 – 2x – 8 over the domain –2 < x < 3
State the absolute and local maximum and minimum values
f ' (x) = 2x – 2
2x – 2 = 0 when x = 1
Turning point is at f(1) = -9
x
y
-2
-1
0
1
2
3
0
-5
-8
-9
-8
-5
absolute maximum is 0
local and absolute minimum
is - 9
4
EXAMPLE 2
Graph the function f(x) = x3 – 9x over the domain –4 < x < 3
State the absolute and local maximum and minimum values
Find the critical values where the slope of the tangent is 0.
f ' (x) = 3x2 – 9
3x2 – 9 = 0
x2 = 3
𝒙 = ± 𝟑 = ±𝟏. 𝟕𝟑
3x2 = 9
5
EXAMPLE 2
Graph the function f(x) = x3 – 9x over the domain –4 < x < 3
State the absolute and local maximum and minimum values
local and absolute max
is 10.4
x
f(x)
-4
-28
-1.73
10.4
1.73
-10.4
3
0
local minimum is -10.4
absolute minimum is -28
6
Homework Assignment
Lesson #2 Maximum and Minimum Values
Complete Questions 1 – 2 on Pages 4 – 6
Check all your work carefully with
your calculator graphs.
7