Calculus Of One Real Variable By Pheng Kim Ving
Chapter 5: Applications Of The Derivative Part 1 Section 5.3: The First-Derivative Test

 

5.3
The First-Derivative Test

 

 

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1. Increasing And Decreasing Functions

 

Consider the function f in Fig. 1.1. As x increases, the graph of f rises, which means that the value f(x) increases. We
say that f is increasing. We note that if x1 < x2, then f(x1) < f(x2).

 

Consider the function f in Fig. 1.2. As x increases, the graph of f falls, which means that the value f(x) decreases. We
say that f is decreasing. We note that if x1 < x2, then f(x1) > f(x2).

 

 

Fig. 1.1

 

f is increasing on [a, b].

 

Fig. 1.2

 

f is decreasing on [a, b].

 

Fig. 1.3

 

f is non-decreasing on [a, b].

 

Fig. 1.4

 

f is non-increasing on [a, b].

 

Definition 1.1

 

 

 

Note that for an increasing function f, f(x) changes in the same direction as x: if x increases then f(x) increases, if x
decreases then f(x) decreases, and that for a decreasing function f, f(x) changes in the opposite direction to x: if x
increases then f(x) decreases, if x decreases then f(x) increases. Also note that the notions of increasing and decreasing
behaviors of f on I don't require the continuity (and thus the differentiability) of f on I.

 

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2. Test For Increasing And Decreasing Behaviors

 

Let the function y = f(x) be differentiable on an interval I. Suppose that f '(x) = dy/dx > 0 for all x in I. Since the rate of
change of f is positive, the change in x and the corresponding change in y must be of the same sign, ie, f must be an
increasing function.

 

Theorem 2.1

 

 

 

Proof

We prove part i. The proofs for all the remaining parts are similar to it. Let x1 and x2 be arbitrary points in I with x1 < x2.
By the mean-value theorem, there exists c in (x1, x2) such that ( f(x2) f(x1))/(x2 x1) = f '(c). Now, c is in I too, so
f '(c) > 0. Also, x2 x1 > 0. Consequently, f(x2) f(x1) > 0, ie, f(x1) < f(x2). Thus f is increasing on I.

EOP

 

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3. Finding Intervals Of Increase And Decrease

 

An interval of increase of f is one where f is increasing. An interval of decrease of f is one where f is decreasing.

 

Example 3.1

 

Find the intervals of increase and decrease of the function f(x) = x3 3x2 + 5.

 

Solution

 

Fig. 3.1

 

Chart conveying increasing and decreasing behavior of f.

EOS

 

Chart Conveying The Increasing And Decreasing Behaviors

 

 

Remark that on an interval, f '(x) > 0 if the number of signs in that interval above the row for f '(x) is 0 or even, and
f '(x) < 0 if that number is odd.

 

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4. Test For Local Extrema

 

As we saw in Section 5.2 Remark 4.2, not every critical point yields a local extremum. We're now going to discuss a test
for determining if a critical point yields a local maximum, or a local minimum, or neither. Recall that local extrema can
occur only at endpoints and critical points.

 

Refer to Fig. 4.1. We have a local maximum of f at x1. We observe that f is increasing on the left of x1 and decreasing on
the right; thus f '(x) > 0 for all points x near and to the left of x1 and f '(x) < 0 for all points x near and to the right of x1.
We also have a local maximum at x3. Both x1 and x3 are critical points: f '(x1) = 0, f '(x3) doesn't exist.

 

Fig. 4.1

 

Local maxima occur at x1 and x3.
Local minima occur at
x2 and x4.
No local extremum at
x5.

 

Theorem 4.1 The First-Derivative Test

 

Suppose that x0 is a critical point of the function f and that f is differentiable on some open interval containing x0 except
possibly at x0 itself, but that f is continuous at x0.

 

i. If f '(x) > 0 on some interval (x1, x0) and f '(x) < 0 on some interval (x0, x2), then f has a local maximum at x0.

 

ii. If f '(x) < 0 on some interval (x1, x0) and f '(x) > 0 on some interval (x0, x2), then f has a local minimum at x0.

 

 

Proof

EOP

 

The test is called a first-derivative test because it involves the first derivative f ' of a function f. Note that it's a test for
local extrema.

 

Remarks 4.1

 

i. f must be differentiable at all nearby points of x0 (this is needed by the conditions in parts i and ii of the theorem). It
may or may not be differentiable at x0. See the points x1, x2, x3, and x4 in Fig. 4.1.

 

ii. f must be continuous at x0. For example, in Fig. 4.2, f is discontinuous at x1, x2, and x3. For each of these points, f is
increasing on some interval to the left of it and decreasing on some interval to the right. Now, f(x1) is a local maximum;

 

Fig. 4.2

 

Discontinuities in this example show that the continuity in the
hypothesis of theorem 2 is essential.

 

f(x2) isn't, because it doesn't exist; and f(x3) isn't, because, even though f(x) < f(x3) for all x to the near right of x3,
f(x) > f(x3) for all x to the near left. So, if f is discontinuous at x0 (and the rest of the hypothesis of the theorem is
satisfied), f(x0) may or may not be a local extremum.

 

iii. If either f '(x) > 0 or f '(x) < 0 on both the near left and the near right of x0, f has no local extremum at x0. For
example, see the point x5 in Fig. 4.1.

 

Test On Endpoints For Local Extrema

 

See Fig. 4.3. Suppose a is the left endpoint of dom( f ) and f is continuous at a and differentiable on the right of a for
some distance. If f '(x) > 0, respectively f '(x) < 0, on some interval (a, x2) (like the case for f1, respectively f2, in Fig.
4.3), then f has a local minimum, respectively a local maximum, at a. Suppose b is the right endpoint of dom( f ) and f
is continuous at b and differentiable on the left of b for some distance. If f '(x) > 0, respectively f '(x) < 0, on some
interval (x1, b) (like the case for f1, respectively f2), then f has a local maximum, respectively a local minimum, at b. The
proof for this result is similar to that of theorem 4.1.

 

Fig. 4.3

 

Local extrema at endpoints.

 

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5. Locating And Classifying Local Extrema

 

Locating local extrema means locating or finding critical points which each yields a local extremum. Classifying a local
extremum means classifying or determining it as either a local maximum or a local minimum.

 

Example 5.1

 

 

Solution

 

The graph of f is sketched in Fig. 5.2.

 

Fig. 5.1

 

f(0) = 1 is neither a local maximum nor a local minimum.
f(3/2) = 11/16 is a local minimum. It's also the absolute minimum.

 

Fig. 5.2

 

Graph of f(x) = x4 2x3 + 1.

EOS

 

Remark 5.1

 

The chart in Fig. 5.1 uses the test for increasing and decreasing behaviors to determine the intervals of increase and
decrease of f. From the chart we determine whether a critical point yields a local extremum or not. The first-derivative
test is implicitly used to confirm that f(3/2) = 11/16 is a local minimum.

 

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Problems & Solutions

 

1. Find the intervals of increase and decrease of the function f(x) = x2 + 2x + 2.

 

Solution

 

 

 

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2. Determine the intervals in which the function y = x4 8x2 + 16 is increasing and decreasing.

 

Solution

 

 

 

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3. Find the intervals of increase and decrease of y = 1/(1 + x2). Find the limits at infinity of y. Sketch the graph of y.

 

Solution

 

We have y' = 2x/(1 + x2)2. So y' = 0 at x = 0. We have y(0) = 1/(1 + 02) = 1. Since the denominator of y' is always
positive, the sign of y' is the same as that of its numerator 2x. The intervals of increase and decrease of y are shown in
the chart below.

 

 

The limits at infinity of y are:

 

 

The graph of y is sketched below.

 

 

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4. For each of the following functions:
- Find limits at infinity and infinite limits,
- Use the first-derivative test to locate critical points,
- Use a test and a chart to determine the intervals of increase and decrease,
- Classify local extreme values if any, and determine if any of them are absolute extreme values.
Then sketch the graph of the function.

 

 

Solution

 

 

so the critical points are x = 1 and x = 1. We have f(1) = (1)/(1 + (1)2) = 1/2 and f(1) = 1/(1 + 12) = 1/2. The
sign of f '(x) is the same as that of the product (1 + x)(1 x).

 

 

Local maximum value: f(1) = 1/2, local minimum value: f(1) = 1/2;
absolute maximum value: f(1) = 1/2, absolute minimum value: f(1) = 1/2.

 

 

 

Note: The vertical double line under a value of x in a row
means that the function in that row isn't defined or doesn't
exist at that value of x.

 

There are no local maximum or minimum values.

 

 

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Solution

 

 

 

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