Minimum and Maximum Values
Section 4.1

Definition of Extrema –
Let f be defined on a interval I
containing c :
f
i. f (c ) is the minimum of
if f (c)  f ( x)
 x I
ii.
f (c )
is the maximum of
if
f (c )  f ( x )
 x I
f
f
on I
on I
Extreme Values (extrema) – minimum
and maximum of a function on an interval
 {can be an interior point or an endpoint}


Referred to as absolute minimum,
absolute maximum and endpoint extrema.
Extreme Value Theorem: {EVT}

If
is continuous on a closed interval
 a, b then f has both a minimum and a
maximum on the interval.

* This theorem tells us only of the
existence of a maximum or minimum
value – it does not tell us how to find it. *
f
Definition of a Relative Extrema:

i. If there is an open interval on which
f (c ) is a maximum, then f (c ) is called a
relative maximum of f . (hill)

ii. If there is an open interval on which
f (c ) is a maximum, then f (c ) is called a
relative minimum of f . (valley)

*** Remember hills and valleys
that are smooth and rounded
have horizontal tangent lines.
Hills and valleys that are
sharp and peaked are not
differentiable at that point!!***
Definition of a Critical Number

If f is defined at c, then c is called a
critical number of f , if f '(c)  0 or if
f '(c)  und .
**Relative Extrema occur only at Critical
Numbers!!**
If f has a relative minimum or relative
maximum at x=c , then c is a critical
number of f.
Guidelines for finding
absolute extrema

i. Find the critical numbers of

ii. Evaluate f at each critical number in
 a, b  .
f
.
y '  0 or y ' 
1
0

iii. Evaluate f at each endpoint  a, b.

iv. The least of these y values is the minimum and
the greatest y value is the maximum.