In the previous section, we saw how derivatives can be used to find rate of change of quantities. In this section, we will see increasing and decreasing functions.
Some basic details can be written in 4 steps:
1. Consider the graph of the function f(x) = x2. It is shown in fig.22.3 below:
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| Fig.22.3 |
2. We see a vertical yellow line. This line helps us to understand the concept of height of graph.
• We see that, the vertical yellow line is drawn through x0.
♦ The lower end of the line is on the x-axis.
♦ The upper end of the line lies on the curve.
• So height of the vertical yellow line is f(x0).
• In such a situation, we say that:
Height of the graph at x0 is f(x0).
• It may be noted that, we do not consider the actual height of the yellow line. We consider the value of f(x0).
♦ Case I: f(x0) = −8
♦ Case II: f(x0) = −3
• Height of the yellow line will be larger in case I. But the value of f(x0) is larger in case II.
3. Now consider the portion of the x-axis from O to X'.
• In this portion, if we move along the x-axis from left to right, we will be passing through points like −3, −2, −1.5, −1, −0.5, −0.25, and finally 0.
• We can find the corresponding f(x) values. They are tabulated in the table 22.1(a) below:
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| Table 22.1(a) |
• The second column of this table, gives the corresponding heights. We see that:
As we move from left to right, the height of the graph decreases.
• In this situation, we write:
The function is decreasing for x < 0
4. Similarly, consider the portion of the x-axis from O to X.
•
In this portion, if we move along the x-axis from left to right, we will be passing through points like 0, 0.25, 0.5, 1, 1.5, 2, 3, . . .
•
We can find the corresponding f(x) values. They are tabulated in the table 22.1(b) above.
•
The second column of this table, gives the corresponding heights. We see that:
As we move from left to right, the height of the graph increases.
•
In this situation, we write:
The function is increasing for x > 0
Based on the above discussion, we can write a definition of increasing and decreasing functions.
Definition 1:
Let I be an open interval contained in the domain of a real valued function f. Then f is said to be
(i) Increasing on I if x1 < x2 in I ⇒ f(x1) ≤ f(x2) for all x1, x2 ∈ I.
(ii) Strictly increasing on I if x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I.
(iii) Decreasing on I if x1 < x2 in I ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ I.
(iv) Strictly decreasing on I if x1 < x2 in I ⇒ f(x1) > f(x2) for all x1, x2 ∈ I.
Now we will write a detailed explanation of the definition. The explanation can be written by using the function f(x) = x2 as an example. It can be written in 6 steps:
1. In the definition, there is a mention about domain of a real valued function.
•
In fig.22.1 above, f(x) = x2 is a real valued function. It's domain is R. So all points in the x-axis, are included in the domain.
2. In the definition, there is a mention about an open interval I.
•
'Open interval' means, the end points should not be used as input values.
•
In our discussion based on fig.22.1 above, we used two open intervals:
♦ (−∞,0) related to the portion OX' of the x-axis.
♦ (0,∞) related to the portion OX of the x-axis.
•
Both of the above intervals are contained in R.
3. Now consider the open interval (−∞,0).
•
Take any two points x1 and x2 such that, x1 < x2.
Let us take x1 = −3 and x2 = −0.25
•
So we get:
♦ f(x1) = f(−3) = (−3)2 = 9
♦ f(x2) = f(−0.25) = (−0.25)2 = 0.0625
•
We can write: f(x1) > f(x2)
•
So item (iv) of the definition is applicable here. We can write:
On the open interval I = (−∞,0), the function f(x) = x2 is strictly decreasing.
•
The general graphical representation of a strictly decreasing function is shown in fig.22.4(d) below:
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| Fig.22.4 |
•
Note that x1, that we select as the first point, should be always less than x2, that we select as the second point. This condition will ensure that, we always move from left to right along the x-axis.
4. Similarly consider the open interval (0,∞).
•
Take any two points x1 and x2 such that, x1 < x2.
Let us take x1 = 0.5 and x2 = 3
•
So we get:
♦ f(x1) = f(0.5) = (0.5)2 = 0.25
♦ f(x2) = f(3) = (3)2 = 9
•
We can write: f(x1) < f(x2)
•
So item (ii) of the definition is applicable here. We can write:
On the open interval I = (0,∞), the function f(x) = x2 is strictly increasing.
•
The general graphical representation of a strictly increasing function is shown in fig.22.4(b) above.
•
Note that x1, that we select as the first point, should be always less
than x2, that we select as the second point. This condition will
ensure that, we always move from left to right along the x-axis.
5. Once we understand strictly decreasing function, we will be able to write about decreasing function. Item (iii) of the definition is applicable here. The only difference between the two can be written as:
♦ For strictly decreasing, we have f(x1) > f(x2)
♦ For decreasing, we have f(x1) ≥ f(x2)
•
That means, for some x1 and x2 values, the corresponding f(x) values may be equal. Such x values will give a horizontal segment to the graph.
•
The general graphical representation of a decreasing function is shown in fig.22.4(c) above.
6. Similarly, once we understand strictly increasing function, we will be able to
write about increasing function. Item (i) of the definition is
applicable here. The only difference between the two can be written as:
♦ For strictly increasing, we have f(x1) < f(x2)
♦ For increasing, we have f(x1) ≤ f(x2)
•
That means, for some x1 and x2 values, the corresponding f(x) values
may be equal. Such x values will give a horizontal segment to the graph.
•
The general graphical representation of a increasing function is shown in fig.22.4(a) above.
Now we will see a solved example.
Solved example 22.7
Show that the function f(x) = 7x − 3 is strictly increasing on R.
Solution:
1. In the discussion above, we considered an interval I inside the domain. In this problem, the interval is same as the domain, which is R.
2. Take any two points x1 and x2 from R. x1 must be less than x2.
•
Let us take x1 = −3 and x2 = 5
3. So we get:
♦ f(x1) = 7(−3) − 3 = −24
♦ f(x2) = 7(5) − 3 = 32
4. We see that, f(x1) < f(x2).
So based on definition 1, we get a hint that, the given f(x) is strictly increasing. However, to prove it beyond doubt, we must write the proof for the general case. It can be done in 3 steps:
1. Take any two points x1 and x2 from R such that, x1 < x2.
2. Since x1 < x2, we get: 7(x1) − 3 < 7(x2) − 3
3. Since, 7(x1) − 3 < 7(x2) − 3, we can write: f(x1) < f(x2)
•
So based on definition 1, we can write: f is a strictly decreasing function.
•
The definition 1 that we saw above, will help us to find the nature of a function within an interval I.
•
The next definition 2 that we are going to see, will help us to find the nature of a function at a point.
We will write the definition first and then see the explanation.
Definition 2:
Let x0 be a point in the domain of a real valued function f. Then f is said to be increasing, strictly increasing, decreasing or strictly decreasing at x0 if there exist an open interval I containing x0 such that f is increasing, strictly increasing, decreasing or strictly decreasing, respectively in I.
The explanation can be written in 3 steps:
1. We are given a real valued function f.
2. x0 is any point in the domain of f.
We want to know the nature of f at the point x0.
3. For that, we consider an open interval:
I = (x0 − h, x0 + h), where h > 0.
•
If I exists such that, f is increasing on I, then we say that f is increasing at x0.
•
If I exists such that, f is strictly increasing on I, then we say that f is strictly increasing at x0.
•
If I exists such that, f is decreasing on I, then we say that f is decreasing at x0.
•
If I exists such that, f is decreasing on I, then we say that f is strictly decreasing at x0.
In the next section, we will see how to apply derivatives to find the nature.
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