Absolute extrema
Local extrema
The closed interval method
§3.1–Maximum and Minimum Values
Tom Lewis
Spring Semester
2014
Absolute extrema
Local extrema
Outline
Absolute extrema
Local extrema
The closed interval method
The closed interval method
Absolute extrema
Local extrema
The closed interval method
Absolute extrema
Local extrema
The closed interval method
Definition
• A function f has an absolute maximum at c if f (c) > f (x ) for
all x ∈ D, where D is the domain of f .
• A function f has an absolute minimum at c if f (c) 6 f (x ) for
all x ∈ D.
• The maximum and minimum values of f are called the
extreme values or extrema of f .
Absolute extrema
Local extrema
The closed interval method
No absolute maximum
The function pictured above has no absolute maximum; it does
have an absolute minimum.
Absolute extrema
Local extrema
The closed interval method
No absolute minimum
The function pictured above has no absolute minimum; it does
have an absolute maximum.
Absolute extrema
Local extrema
The closed interval method
No extrema
The function pictured above has no extreme values.
Absolute extrema
Local extrema
The closed interval method
Theorem (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].
Problem
Review the last three examples. Why doesn’t the Extreme Value
Theorem apply to them?
Absolute extrema
Local extrema
The closed interval method
Absolute extrema
Local extrema
The closed interval method
Definition
• A function f has a local maximum (or relative maximum) at c
if there exists an open interval I containing c such that
f (c) > f (x ) for all x ∈ I .
• A function f has a local minimum (or relative minimum) at c
if there exists an open interval I containing c such that
f (c) 6 f (x ) for all x ∈ I .
• The local minimum or local maximum values of a function are
collectively called local extreme values (or relative extreme
values).
Observe
According to this definition, an endpoint can never be the location
of a local extrema. Why?
Absolute extrema
Local extrema
The closed interval method
Theorem (Fermat’s Theorem)
If f has a relative extreme value at c, and if f 0 (c) exists, then
f 0 (c) = 0.
Absolute extrema
Local extrema
The closed interval method
Two things to remember
• The converse of Fermat’s theorem (if f 0 (c) = 0, then f has a
local extreme value at c) is false. Consider f (x ) = x 3 and
c = 0. Clearly f 0 (0) = 0, but f does not have a local extreme
value at 0.
• A function may have local extreme values at points where f is
non-differentiable. The function f (x ) = |x | has a local
minimum value at c = 0, but f 0 (0) does not exist.
Absolute extrema
Local extrema
The closed interval method
Definition
A critical number of a function f is a number c in the domain of f
such that either
1. f 0 (c) = 0 or
2. f 0 (c) does not exist.
Theorem
If f has a local extreme value at c, then c is a critical number of f .
Absolute extrema
Local extrema
The closed interval method
Problem
Find the critical numbers of f (x ) = x 2/5 (2 − x ) on R.
Absolute extrema
Local extrema
The closed interval method
Problem
Find the critical numbers of f (θ) = θ − 2 cos(θ) on the interval
(0, 2π).
Absolute extrema
Local extrema
The closed interval method
Theorem
If f is continuous on a closed interval [a, b], then the absolute
extrema of f must be located among the following points:
• The critical numbers of f ;
• The endpoints of the interval, {a, b}.
Absolute extrema
Local extrema
The closed interval method
Problem
Find the extreme values of f (x ) = x 2 (3 − x ) on the interval [−1, 4].