MAT1234 Handout 3.1
1. A function f has an absolute/global 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/global 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.
2. 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 1
DO NOT SKIP STEPS.
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3. The Extreme Value Theorem
If f is continuous on  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 .
Example 2
f is not continuous on [a,b]
The interval is not closed
4. Fermat’s Theorem
If f has a local maximum or minimum at c , and if f (c ) exists, then f (c)  0 .
T or F: If f (c)  0 , then f has a local maximum or minimum at c.
DO NOT SKIP STEPS.
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6. A critical number of a function f is a number c in the domain of f such that either
f (c)  0 or f (c ) does not exist.
Example 3 Find all the critical numbers of f ( x)  5  6 x  2 x3 .
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 the those values from is the absolute minimum value.
DO NOT SKIP STEPS.
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Example 4 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






DO NOT SKIP STEPS.
Pay attention to the Presentations
You need to give formal
conclusions to the problem.
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Classwork
1. Find the absolute maximum and absolute minimum values of f on the given interval.
f ( x)  18x  15x 2  4 x3 , [  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






DO NOT SKIP STEPS.
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