MAT1221 Section 3.2 Handout
Absolute Maximum
A function f has an absolute maximum at c if f (c)  f ( x) for all x in D, where D is the
domain of f. The number f(c) is called the maximum value of f on D.
(A function f has an absolute minimum at c if f (c)  f ( x) for all x in D, where D is the domain of f.
The number f(c) is called the minimum value of f on D.)
The absolute maximum and minimum values of f are called the extreme values of f.
Local Maximum
A function f has a local/relative maximum at c if f (c)  f ( x) when x is near c.
[This means that f (c)  f ( x) for all x in some open interval containing c.]
(A function f has a local/relative minimum at c if f (c)  f ( x) when x is near c.)
Example 0
Note An end point is not consider as a local max/min.
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The First Derivative Test
Suppose that c is a critical number of a continuous function f.
(a) If f  changes from positive to negative at c, then f has a local maximum at c.
(b) If f  changes from negative to positive at c, then f has a local minimum at c.
(c) If
f  does not changes sign at c, then f has no local max. or min. at c.
Example 1 (Continue from 3.1) f ( x)  x 2  6 x  10
The local minimum value of f is
Expectations
You are expected to give formal
conclusions to the problem with
specific information.
The Extreme value Theorem
If f is continuous on a closed interval  a, b , then f attains an absolute maximum value
f  c  and an absolute minimum value f  d  at some numbers c and d in  a, b .
The Closed Interval Method To find the absolute maximum and minimum values of a
continuous function f on a closed interval  a, b :
1. Find the values of f at the critical numbers of f in  a, b  .
2. Find the values of f at the end points.
3. The largest of the values from steps 1 and 2 is the absolute maximum value;
the smallest of those values is the absolute minimum value.
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Example 1 Find the absolute maximum and absolute minimum values of
f ( x)  x3  12 x  1 on [3,5] .
Step 1: Find the critical numbers of f on (-3,5).
Step 2: Compare the function values at the critical numbers and the end points of the
interval.
Step 3: Make formal conclusions.
The absolute maximum value of f is

f






The absolute minimum value of f is

f






Expectations
You are expected to give formal
conclusions to the problem with
specific information.
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Classwork Find the absolute maximum and absolute minimum values of
f ( x)  18x  15x 2  4 x3 on [3, 4] .
Step 1: Find the critical numbers of f on (-3,4).
Step 2: Compare the function values at the critical numbers and the end points of the
interval.
Step 3: Make formal conclusions.
The absolute maximum value of f is

f






The absolute minimum value of f is

f






Expectations
You are expected to give formal
conclusions to the problem with
specific information.
Additional HW #23
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