Page 54 - Mathematics Class - XII

3. Take another two points say P and P on the same curve, and make tangents, using the same wire, at P
2
2
3
and P making angles a and a , respectively with the positive direction of x-axis.
3 2 3
4. Here again a and a are obtuse angles and therefore slopes of the tangents tan a and tan a are both
2
2
3
3
negative, i.e., derivatives of the function at P and P are negative.
3
2
5. Hence, the function given by the curve (on the left) is a decreasing function.
6. Again, on the curve (on the right), take three point Q , Q , Q and using the other straight wires, form
3
1
2
tangents at each of these points making angles b , b , b respectively with the positive direction of x-axis,
3
2
1
as shown in the Fig. (b).
7. Here, b , b , b are all acute angles. So, the derivatives of the function at these points are positive. Thus,
1
3
2
the function given by this curve (on the right) is an increasing function.
OBSERVATION
We observe that
1. a = 110° > 90°, a = 120° > 90°, a = 130° > 90°, all are obtuse angles, therefore slope of tangent at P ,
1
2
1
3
P , P are negative.
3
2
2. tan a = tan 110° = – 2.7475 (negative)
1
tan a = tan 120° = – 1.7321 (negative)
2
tan a = tan 130° = – 1.1918 (negative)
3
So, the given by curve (on the left) is a decreasing function.
3. Now, on measuring b = 70° < 90°, b = 60° < 90°, b = 50° < 90°, all are acute angles, therefore slope of
1
3
2
tangent at Q , Q , Q are positive.
1 2 3
tan b = tan 70° = + 2.7475 (positive)
1
tan b = tan 60° = + 1.7321 (positive)
2
tan b = tan 50° = + 1.1918 (positive)
3
Thus, the function given by the curve (on the right) is an increasing function.

CONCLUSION
From this activity, it is verified that a function f (x) is increasing when f ′(x) ≥ 0 and decreasing when
f ′(x) ≤ 0 (where x ∈ Domain f ).

APPLICATION
This activity may be useful in explaining the concepts of decreasing and increasing functions.

Knowledge Booster
A function f (x) is said to be strictly increasing on an interval I. If f (b) > f (a) for all b > a, where a, b in I.
Conversely,
A function f (x) decreases on an interval I if f (b) ≤ f (a) for all b > a with a, b in I. If f (b) < f (a) for all b > a,
the function is said to be strictly decreasing.
On the other hand, if f (b) ≥ f (a) for all b > a, the function is said to be (non-strictly) increasing.

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