Calculus
Of One Real Variable – By Pheng Kim Ving 
5.2 
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1. Maxima, Minima, And Extrema 
We see in Section
1.2.2 Theorem 2.1 that if a function f is
continuous on the closed finite interval [a, b] then f attains
both a maximum and a minimum on [a, b]. However we haven't yet discussed a way to find the
maximum and the
minimum of such a function. Functions that aren't continuous on closed finite intervals
of the form [a, b]
may or may not
attain a maximum or a minimum.
In Section
5.1 Definitions 1.1 we have the definitions of local maximum and local
minimum. Recall that the maximum and
minimum of f are also called absolute maximum
and absolute minimum respectively, because they're the maximum
and minimum of f respectively on the entire domain of f;
that extremum is a maximum or a minimum; and that
maxima
and minima are collectively referred to as extrema. An absolute
extremum is an absolute maximum or an absoute
minimum, and absolute extrema are absolute maximum and absolute minimum.
The objective of this section is
to investigate ways to find the absolute maximum and minimum, if any, of
functions
continuous on a closed finite interval of the form [a,
b] and also of some other functions.
An absolute maximum or minimum may
occur at one or more points. For example, in Fig. 1.1, the absolute maximum of
f occurs at two points: x_{1} and x_{2}; the absolute
minimum of f occurs at only one point: a.
Fig. 1.1
An absolute maximum or minimum may occur
at one or more

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2. Absolute Extrema Of Functions Continuous On [a, b] 
Let's find absolute extrema of a function f continuous on a closed finite interval [a, b]. Since an
absolute extremum is
also a local one, the absolute extrema must be among the local ones. Now, an
endpoint of dom( f ) may or
may not yield
a local extremum. It may seem that every endpoint must yield a local extremum.
But that's
not so. This fact is difficult to
see, and examples proving it are rare, but they exist. When an endpoint yields
a local extremum, that local extremum, like
all local extrema, may also be the absolute one; this is the case of the local
and absolute minimum f(a) in Fig. 1.1. Thus,
the absolute extrema must be among f(a), f(b), and the local extrema in between.
We clearly see that the absolute
maximum of f is the largest of f(a), f(b), and the
local extrema, and that the absolute
minimum of f is the smallest of them.
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3. Places Where The Local Extrema Can Be 
At Points x Where
f '(x) = 0
The discussion in this part applies
to functions in general situation, not just the ones continuous on closed
finite intervals
of the form [a, b].
As seen in Section
5.1 Theorem 2.1, if f(x_{1}) is a local extremum and if f '(x_{1}) exists, then f '(x_{1}) = 0.
The graph of g(x) = x^{2} is sketched in
Fig. 3.1. We have g'(0) = g'(x)_{x}_{=}_{0} = 2x_{x}_{=}_{0} = 2(0) = 0,
and g(0) = 0 is a local
minimum. The graph of h(x) = x^{3} is sketched in
Fig. 3.2. We have h'(0) = h'(x)_{x}_{=}_{0} = 3x^{2}_{x}_{=}_{0} = 3(0^{2})
= 0, but h(0) = 0 is
neither a local maximum nor a local minimum.
Hence, at a point x where f '(x) = 0, a local extremum may or may not occur. Although
local extrema can occur at points
x where f '(x) = 0, they don't have to occur at such points.
Fig. 3.1 y = g(x) = x^{2}; 
Fig. 3.2 y = h(x) = x^{3}; 
At Points x Where
f '(x) Doesn't Exist
Now, at a point x_{1} where f(x_{1}) is a local extremum, must f
always be differentiable?
Can f be nondifferentiable at x_{1}?
Let's take
an example. Consider g(x) = x^{2/3}. See Fig.
3.3. We have g'(x) = (2/3)x^{–1/3} = 2/(3x^{1/3}). So g'(0)
doesn't
exist.
And g(0) = 0 is a local minimum.
Therefore, f can be nondifferentiable at a
point x_{1} where f(x_{1}) is a local minimum and
in general a local extremum. Consequently, the second and last place to look
for local extrema, a place other than the
endpoints, is the set of all the points x where f is nondifferentiable, ie, where f '(x) doesn't exist.
Like the situation with the points
x where f '(x) = 0, a local extremum may or may not occur at
a point x where f
'(x)
doesn't
exist. Examples are illustrated in Figs. 3.3 and 3.4.
Fig. 3.3 y = g(x) = x^{2/3};

Fig. 3.4 Graph of y = h(x) exhibits a sharp point at x = x_{1}, so h'(x_{1}) doesn't exist;

In summary:
The places where to look for
local extrema of f are the
endpoints of dom( f ) and the
points x of dom ( f ) where 
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4. Critical Points 
A point x_{1} in the domain
of a function f and not being an endpoint
of the domain is called a critical point of f
if either 
i. We exclude the endpoints because otherwise they
would always be critical points due to the fact that f isn't
differentiable
at endpoints.
ii. If f '(x_{1}) doesn't exist, then x_{1} is also called a singular point of f.
iii. The definition applies to functions in general
situation, not just functions continuous on closed finite intervals of the
form [a,
b] where a and b are finite numbers.
So:
Local extrema of f can occur
only at endpoints and critical points of f. However,
not every endpoint or critical point 
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5. Finding Absolute Extrema Of Functions Continuous On [a, b] 
Again recall from Section
1.2.2 Theorem 2.1 that if a function is continuous on a closed finite interval [a, b], then it
attains both an absolute maximum and an absolute minimum there.
Here we summarize all that has
been discussed above. To find the absolute extrema of a function f continuous
on the
closed finite interval [a, b], we proceed in three steps as follows:
i. Find all critical points of f in (a, b).
ii. Compute f(a), f(b), and the values of f at all the critical points.
iii. Among the values obtained in part ii, the
greatest is the absolute maximum of f and the
least is the absolute
minimum of f.
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6. Absolute Extrema Of Discontinuous Functions 
Consider the function f which is discontinuous at x
= a and continuous at all other points
in [a, b],
ie, continuous at all
points in (a, b].
See Figs. 6.1 thru 6.4.
In Fig. 6.2, f
isn't
defined at a but is bounded near it. We can make
f(x) get as
close to v as we please, but f(x) never is
v because a doesn't belong to dom ( f ). Thus, v cannot be
the absolute maximum of f (recall: a
maximum or minimum,
absolute or local, of f is a value of f ).
Does f has an absolute maximum? No, because there's no “ first point ” to the
right of a: if x_{1} is a point to
the right of and close to a, then x_{2} =
a + (x_{1} – a)/2 = (a + x_{1})/2 is to the right of a
and
closer to it than x_{1} is. The absolute minimum of f is f(m), attained at x
= m.
Fig. 6.1 Absolute maximum of f: none.

Fig. 6.2 Absolute maximum of f: none.

Fig. 6.3 Absolute maximum of f: none.

Fig. 6.4 Absolute maximum of f: f(a).

In Fig. 6.3, f
is defined at a, but the value f(a) of f at a is smaller
than v. Consequently, as in Fig. 6.2, f has no absolute
maximum. Again the absolute minimum of f is f(m), attained
at x = m
(note that f(a) > f(m)).
In Fig. 6.4, f
is defined at a, and f(a) > v. Now f has an absolute maximum, wow! It's f(a), attained
at x = a.
The
absolute minimum of f is, well,
still f(m), attained
at x = m.
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7. Absolute Extrema Of Functions On Unbounded
Intervals 
Fig. 7.1 Absolute maximum of f: none.

Fig. 7.2 Absolute maximum of f: none.

Fig. 7.3 Absolute maximum of f: f(a).

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8. Remark 
Parts 6 and
7 show that to find the absolute extrema of a function f that is discontinuous and/or has an unbounded
Problems & Solutions 
1. Sketch the graph of y
= f(x) = x^{2}. Find the absolute extrema of f
if any on:
a. [–2, 1].
b. (–2, 1).
c. [1, 3].
d. (1, 3].
Solution
a. f '(x) = 2x, so f '(x) exists everywhere and f
'(x) = 0 only at x
= 0. As 0 is in [–2, 1] and f '(x) is defined for all x
in
[–2, 1], there's only one
critical point: x = 0. We have:
f(–2) = 4,
f(1)
= 1, and
f(0)
= 0.
Thus the absolute maximum of f
on [–2, 1] is 4 attained at x = –2 and
its absolute minimum there is 0 attained at x
= 0.
This shows that on (1, 3], f has an absolute maximum of 9 attained at x = 3 but it has no absolute minimum.
2. Find the absolute extrema of y = f(x) = x^{2} if any on:
Solution
3. Find the absolute extrema of y = f(x) = x^{2} if any on:
Solution
Consequently on S_{2}, f has no absolute maximum but it has an absolute minimum of 4 attained at x = –2.
Solution
5. Let a < b < c. Suppose a
function f
is continuous on [a,
c]
and f
'(x) > 0 on
(a, b) and (b, c). Prove
that f(b) is
neither a local maximum nor a local
minimum of f.
Solution
A similar argument will lead to the fact that f(b) < f(x) for all x in (b, c], and completes the proof.
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