Calculus Of One Real Variable By Pheng Kim Ving
Chapter 5: Applications Of The Derivative Part 1 Section 5.2: Critical Points and Extrema

 

5.2
Critical Points And Extrema

 

 

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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: x1 and x2; 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
points.

 

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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(x1) is a local extremum and if f '(x1) exists, then f '(x1) = 0.

 

The graph of g(x) = x2 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) = x3 is sketched in Fig. 3.2. We have h'(0) = h'(x)|x=0 = 3x2|x=0 = 3(02) = 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) = x2;
g'(0) = 0;
g(0) = 0 is a local minimum.

 

Fig. 3.2

 

y = h(x) = x3;
h'(0) = 0;
h(0) = 0 is neither a local maximum nor a local minimum.

 

At Points x Where f '(x) Doesn't Exist

 

Now, at a point x1 where f(x1) is a local extremum, must f always be differentiable? Can f be non-differentiable at x1?
Let's take an example. Consider g(x) = x2/3. See Fig. 3.3. We have g'(x) = (2/3)x1/3 = 2/(3x1/3). So g'(0) doesn't exist.
And g(0) = 0 is a local minimum. Therefore, f can be non-differentiable at a point x1 where f(x1) 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 non-differentiable, 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) = x2/3;
g'(0) doesn't exist;
g(0) is a local minimum.

 

Fig. 3.4

 

Graph of y = h(x) exhibits a sharp point at x = x1, so h'(x1) doesn't exist;
h(x1) is neither a local maximum nor a local minimum.

 

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
either f '(x) = 0 or f '(x) doesn't exist.

 

 

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4. Critical Points

 

Definition 4.1

 

A point x1 in the domain of a function f and not being an endpoint of the domain is called a critical point of f if either
f '(x1) = 0 or f '(x1) doesn't exist.

 

 

Remarks 4.1

 

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 '(x1) doesn't exist, then x1 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.

 

Remark 4.2

 

So:

 

 

Local extrema of f can occur only at endpoints and critical points of f. However, not every endpoint or critical point
yields a local extremum.

 

 

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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 x1 is a point to the right of and close to a, then x2 = a + (x1 a)/2 = (a + x1)/2 is to the right of a and
closer to it than x1 is. The absolute minimum of f is f(m), attained at x = m.

 

Fig. 6.1

 

Absolute maximum of f: none.
Absolute minimum of
f: f(m).

 

Fig. 6.2

 

Absolute maximum of f: none.
Absolute minimum of
f: f(m).

 

Fig. 6.3

 

Absolute maximum of f: none.
Absolute minimum of
f: f(m).

 

Fig. 6.4

 

Absolute maximum of f: f(a).
Absolute minimum of
f: f(m).

 

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.
Absolute minimum of
f: f(m).

 

Fig. 7.2

 

Absolute maximum of f: none.
Absolute minimum of
f: f(m).

 

Fig. 7.3

 

Absolute maximum of f: f(a).
Absolute minimum of
f: f(m).

 

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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

 

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Problems & Solutions

 

1. Sketch the graph of y = f(x) = x2. 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.

 

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2. Find the absolute extrema of y = f(x) = x2 if any on:



Solution

 

 

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3. Find the absolute extrema of y = f(x) = x2 if any on:



Solution

 

 

Consequently on S2, f has no absolute maximum but it has an absolute minimum of 4 attained at x = 2.

 

 

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Solution

 

 

 

 

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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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