Math 2413
Notes 3.4
Section 3.4 – Extreme Values
Definition:
A function f is said to take on a local maximum at c if:
f (c)  f ( x) for all x sufficiently close to c.
A function f is said to take on a local minimum at c if:
f (c)  f ( x ) for all x sufficiently close to c.
Theorem:
If f takes on a local maximum or minimum at c, then either
f ' (c)  0 or f ' (c ) does not exist.
Definition:
The numbers c in the domain of a function f for which either f ' ( x )  0 or f ' (c) does not exist,
are called the critical numbers of f.
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Math 2413
Notes 3.4
Theorem: The First Derivative Test
Suppose that c is a critical number of f and f is continuous at c. If there is a positive number δ such that:
i.
f ' ( x )  0 for all x in (c   , c) and f ' ( x)  0 for all x in (c, c   ) then f (c) is a local maximum.
ii.
f ' ( x )  0 for all x in (c   , c) and f ' ( x)  0 for all x in (c, c   ) then f (c) is a local minimum.
iii.
f ' ( x ) keeps constant sign on for all x in (c   , c)  (c, c   ) then f (c ) is NOT a local extreme value.
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Math 2413
Notes 3.4
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Math 2413
Notes 3.4
Examples:
1. The graph of f ′ (x) ,the derivative of f(x), is shown below.
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Math 2413
Notes 3.4
2.
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Math 2413
Notes 3.4
3
Example 1: Find all critical numbers and the local extreme values of function f ( x )  x  3x.
Example 2: Find all critical numbers and the local extreme values of function f ( x )  x 3  3 x  2 .
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Math 2413
Notes 3.4
Example 3: Find all critical numbers and the local extreme values of function f ( x )  x 
Example 4: Find all critical numbers and the local extreme values of function f ( x) 
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1
.
x
1 x
.
1 x
Math 2413
Notes 3.4
2
Example 5: Find all critical numbers and the local extreme values of function f ( x ) 
x
.
1 x
2
1
Example 6: Find all critical numbers and the local extreme values of function f ( x )  x 3  2 x 3 .
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Math 2413
Notes 3.4
Example 8: Find all critical numbers and the local extreme values of function
f ( x)  sin x  cos x on 0  x  2 .
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Math 2413
Notes 3.4
Example 9: Find all critical numbers and the local extreme values of function
f ( x)  x  cos(2 x) on 0  x   .
Example 10: Find all critical numbers and the local extreme values of function f ( x )  sin 2 x on 0  x  2 .
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