AP Calculus Section 3.1
Extrema on an Interval
Definitions, Theorems, and Examples
Definition
Let f be defined on an interval I containing c.
1.) f(c) is a minimum of f on I if f(c) ≤ f(x) for all x
in I.
2.) f(c) is a maximum of f on I if f(c) ≥ f(x) for all x
in I.
Extrema
The minimum and maximum of a function on an
interval are the extreme values or extrema
(singular extremum), of the function on the
interval. The minimum and maximum are also
called the absolute minimum and absolute
maximum on the interval.
A function does not have to have a minimum or
a maximum on an interval.
Extrema can occur at interior points or
endpoints of an interval. Extrema that
occur at the endpoints are called
endpoint extrema.
Theorem 3.1
The Extreme Value Theorem
If f is continuous on the closed interval [a, b],
then f has both a minimum and a maximum on
the interval.
Definition
1.) If there is an open interval containing c on
which f(c) is a maximum, then f(c) is called a
relative maximum of f. f has a relative maximum
at (c, f(c)).
2.) If there is an open interval containing c on
which f(c) is a minimum, then f(c) is called a
relative minimum. f has a relative minimum at
(c, f(c)).
Example 1
Find the value of the derivative at each of the
relative extrema of
A.) 𝑓 𝑥 =
4𝑥
𝑥 2 +1
Find the value of the derivative at each of the
relative extrema of
B.) 𝑔 𝑥 = 𝑥 − 3
Find the value of the derivative at each of the
relative extrema of
C.) ℎ 𝑥 = cos 𝑥
Definition
Let f be defined at c. If f’(c) = 0
or f is not differentiable at c,
then c is a critical number of f.
Theorem 3.2
If f has a relative maximum or
relative minimum at x = c, then
c is a critical number of f.
Guidelines for Finding Extrema on a
Closed Interval
f is a continuous function on [a, b].
1.) Find the critical numbers of f in (a, b).
2.) Evaluate f at each critical number.
3.) Evaluate f at each endpoint of [a, b].
4.) The least of these values is the
minimum. The greatest is the maximum.
Example 2 (#23)
Find the extrema and critical numbers of 𝑓 𝑥 =
3 2
3
𝑥 − 𝑥 on [-1, 2].
2
Note: On Example 2, the critical
numbers x = 0 and x = 1 do not yield a
maximum or a minimum. This tells us
that the converse of Theorem 3.2 is
not true.
THE CRITICAL NUMBERS OF A
FUNCTION DO NOT HAVE TO PRODUCE
RELATIVE EXTREMA.
Example 3 (#25)
Find the extrema and critical numbers of 𝑦 =
2
3
3𝑥 − 2x on [-1, 1].
Example 4 (#17)
Find the extrema and critical numbers of ℎ 𝑥 =
𝑠𝑖𝑛2 𝑥 + cos 𝑥 on [0, 2𝜋]
Assignment: Section 3.1
p.169-170 #2, 4, 8, 18, 24, 26, 30, 34,
46