4'1 Maximum and Minimum Values

1
4.1 Maximum and Minimum Values
2
Definition
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 maximum and minimum values are called extreme
values.
3
Absolute Maximum and Minimum
Absolute Maximum
Absolute Minimum
4
Definition
A function f has a local (relative) maximum at c
if f (c) f (x) where x is in a small
open interval about c.
A function f has a local (relative) minimum at c
if f (c) f (x) where x is in a small
open interval about c.
5
Absolute Maximum and Minimum
Relative Maximum
Relative Minimum
6
The Extreme Value Theorem
If f is continuous on a closed interval a, b,
then f attains an absolute maximum value f (c)
and absolute minimum f (d) at some numbers c and
d in a, b.
Note Absolute maximums and minimums can occur at
endpoints, but local maximums and minimums cannot.
7
Absolute Maximum and Minimum
Absolute Maximum


Absolute Minimum
8
Notes on Extreme Values
Note Absolute maximums and minimums can occur at
endpoints, but local maximums and minimums cannot.
Note On a closed interval, a function can have
multiple absolute and local extreme values.
9
Fermats Theorem
Question If a maximum or minimum occurs at c,
then what is the value of f '(c)?
If f has a local maximum or minimum at c, and if
f '(c) exists, then f '(c) 0.
10
Critical Number or Value
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.
Question If f '(c) 0, do we necessarily have
a maximum or minimum at x c. That is, does
every critical number result in a maximum or
minimum?
11
Absolute Maximum and Minimum
No extreme value at c1.

c2
c1
12
Critical Number or Value
Question If f '(c) 0, do we necessarily have
a maximum or minimum at x c. That is, does
every critical number result in a maximum or
minimum?
Answer If f '(c) 0 does not guarantee that we
have a maximum or minimum at x c.
If f has a local maximum or minimum at c, then c
is a critical number of f, that is f '(c) 0.
The reverse is not necessarily true.
13
The Closed Interval Method
To find an 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 number of
f in (a, b).
2. Find the values of f at the end points of the
interval.
3. The largest of the values in Steps 1 and 2 is
the absolute maximum value and the smallest is
the absolute minimum value.