Section 5.2: Maximum and Minimum Values
Definition: Let f be a function with domain D that contains the number c.
• The function has an absolute maximum at c if f (c) ≥ f (x) for all x ∈ D. The
number f (c) is called the absolute maximum value of f on D. Similarly, f has an
absolute minimum at c if f (c) ≤ f (x) for all x ∈ D and the number f (c) is called
the absolute minimum value of f on D. The absolute maximum and minimum
values of f are called the absolute extrema of f .
• The function has a local maximum at c if f (c) ≥ f (x) for all x in some open interval
containing c. The number f (c) is called a local maximum value of f . Similarly, f
has a local minimum at c if f (c) ≤ f (x) for all x in some open interval containing
c and the number f (c) is called a local minimum value of f . The local maximum
and minimum values of f are called the local extrema of f .
Example: Find the absolute and local extrema of each function by graphing.
(a) f (x) = 1 − x2 , −2 ≤ x ≤ 1
1
(b) f (x) =
1
,0<x≤1
x
(c) f (x) = cos x, −2π ≤ x ≤ 2π
(d) f (x) =
if −1 ≤ x < 0
x2
2 − x2 if 0 ≤ x ≤ 1.
2
Theorem: (Fermat’s Theorem)
If f has a local extremum at c and f 0 (c) exists, then f 0 (c) = 0.
Note: If f 0 (c) = 0, f does not necessarily have an extremum at c. For instance, if f (x) = x3 ,
then f 0 (0) = 0 but f has no maximum or minimum at x = 0.
Similarly, if f has an extremum at c, then f 0 may not exist. For example, f (x) = |x| has an
absolute minimum at x = 0, but f 0 (0) does not exist.
Definition: A critical number of a function f is a number c in the domain of f such that
either f 0 (c) = 0 or f 0 (c) does not exist.
Example: Find the critical numbers of each function.
(a) f (x) = 4x3 − 9x2 − 12x + 3
3
(b) g(x) = |4 − x2 |
(c) f (x) =
(d) g(x) =
x2
√
x
+1
x2 − x
4
Theorem: (Extreme Value Theorem)
If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and
absolute minimum value on [a, b].
The Closed Interval Method: To find the absolute extrema of a continuous function f on a
closed interval [a, b]:
1. Find the values of f at each of the critical numbers in (a, b).
2. Find the values of f at the endpoints x = a and x = b.
3. The largest of these values is the absolute maximum value and the smallest is the
absolute minimum value.
Example: Find the absolute extrema of each function on the given interval.
(a) f (x) = x3 − 12x + 1, [−3, 5]
(b) f (x) = 3x5 − 5x3 − 1, [−2, 2]
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(c) f (x) = x − 2 cos x, [−π, π]
(d) f (x) = xe−x , [0, 2]
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