We
begin
with
a
definition
which
is
illustrated
in
Figure
4.
DEFINITION
Let
/
be
a
function,
and
A
a
set
of
numbers
contained
in
the
domain
of
/.
A
point
x
in
A
is
a
local
maximum
[minimum]
point
for
/
on
A
if
there
is
some
<5
>
such
that
x
is
a
maximum
[minimum]
point
for
/
on
A nix
8,x
+
8).
THEOREM
2
If
x
is
a
local
maximum
or
minimum
for
/
on
(a,
b)
and
/
is
differentiable
at
x,
then
fix)
=
0.
proof
You
should
see
why
this
is
an
easy
application
of
Theorem
1
.

1
90
Derivatives
and
Integrals
The
converse
of
Theorem
2
is
definitely
not
true
—
it
is
possible
for
f'(x)
to
be
even
if
x
is
not
a
local
maximum
or
minimum
point
for
/.
The
simplest
example
is
provided
by
the
function
f(x)
—
x
3
;
in
this
case
f'(0)
=
0,
but
/
has
no
local
maximum
or
minimum
anywhere.
Probably
the
most
widespread
misconceptions
about
calculus
are
concerned
with
the
behavior
of
a
function
/
near
x
when
f'(x)
=
0.
The
point
made
in
the
previous
paragraph
is
so
quickly
forgotten
by
those
who
want
the
world
to
be
simpler
than
it
is,
that
we
will
repeat
it:
the
converse
of
Theorem
2
is
not
true
—
the
condition
f'(x)
—
does
not
imply
that
x
is
a
local
maximum
or
minimum
point
of
/.
Precisely
for
this
reason,
special
terminology
has
been
adopted
to
describe
numbers
x
which
satisfy
the
condition
fix)
—
0.
DEFINITION
A
critical
point
of
a
function
/
is
a
number
x
such
that
f'(x)
=
0.
The
number
f(x)
itself
is
called
a
critical
value
of
/.
x\
a
local
minimum
point
FIG
1
RE
4
X2
a
local
maximum
point
The
critical
values
of
/,
together
with
a
few
other
numbers,
turn
out
to
be
the
ones
which
must
be
considered
in
order
to
find
the
maximum
and
minimum
of
a
given
function
/.
To
the
uninitiated,
finding
the
maximum
and
minimum
value
of
a
function
represents
one
of
the
most
intriguing
aspects
of
calculus,
and
there
is
no
denying
that
problems
of
this
sort
are
fun
(until
you
have
done
your
first
hundred
or
so).
Let
us
consider
first
the
problem
of
finding
the
maximum
or
minimum
of
/
on
a
closed
interval
[a,/?].
(Then,
if
/
is
continuous,
we
can
at
least
be
sure
that
a
maximum
and
minimum
value
exist.)
In
order
to
locate
the
maximum
and
minimum
of
/
three
kinds
of
points
must
be
considered:
(1)
The
critical
points
of
/
in
[a,
b].
(2)
The
end
points
a
and
b.
(3)
Points
x
in
[a,
b]
such
that
/
is
not
differentiable
at
x.
If
x
is
a
maximum
point
or
a
minimum
point
for
/
on
[a,
£>]
,
then
x
must
be
in
one
of
the
three
classes
listed
above:
for
if
x
is
not
in
the
second
or
third
group,
then
x
is
in
(a,
b)
and
/
is
differentiable
at
x;
consequentiy
f'(x)
=
0,
by
Theorem
1,
and
this
means
that
.v
is
in
the
first
group.
If
there
are
many
points
in
these
three
categories,
finding
the
maximum
and
minimum
of
/
may
still
be
a
hopeless
proposition,
but
when
there
are
only
a
few
critical
points,
and
only
a
few
points
where
/
is
not
differentiable,
the
procedure
is
fairly
straightforward:
one
simply
finds
f(x)
for
each x
satisfying
f'(x)
=
0,
and
f(x)
for
each
x
such
that
/
is
not
differentiable
at
x
and,
finally,
f{a)
and
fib).