Chapter 17 Increasing and Decreasing Functions Ex. 17.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 17 Increasing and Decreasing Functions Ex. 17.2

Question 1(i)

Find the intervals in which the following functions are increasing or decreasing:

10 – 6x – 2x²Solution 1(i)

Question 1(ii)

Find the intervals in which the following functions are increasing or decreasing:

x² + 2x – 5Solution 1(ii)

Question 1(iii)

Find the intervals in which the following functions are increasing or decreasing:

6 – 9x – x²Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 1(xiii)

Solution 1(xiii)

Question 1(xiv)

Solution 1(xiv)

Question 1(xv)

Solution 1(xv)

Question 1(xvi)

Solution 1(xvi)

Question 1(xvii)

Solution 1(xvii)

Question 1(xviii)

Solution 1(xviii)

Question 1(xix)

Solution 1(xix)

Question 1(xx)

Solution 1(xx)

Question 1(xxi)

Solution 1(xxi)

Question 1(xxii)

Solution 1(xxii)

Question 1(xxiii)

Solution 1(xxiii)

Question 1(xxiv)

Solution 1(xxiv)

Question 1(xxv)

Find the values of x for which the function y = [x(x – 2)]² is increasing or decreasingSolution 1(xxv)

Question 1(xxvi)

Find the interval in which the following function is increasing or decreasing.

f(x) = 3x⁴– 4x³– 12x² + 5Solution 1(xxvi)

Question 1(xxvii)

Find the interval in which the following function is increasing or decreasing.

Solution 1(xxvii)

Question 1(xxviii)

Find the interval in which the following function is increasing or decreasing.

Solution 1(xxviii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Show that the function given by f(x) = sin x is

(a) increasing in (0, π/2)

(b) decreasing in (π/2, π)

(c) neither increasing nor decreasing in (0, π)Solution 7

Question 8

Prove that the function f given by f(x) = log sin x is increasing on  and decreasing on Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30(i)

Solution 30(i)

Question 31

Solution 31

Question 32

Solution 32

Question 33

Prove that the function f(x) = cos x is:

(i) strictly decreasing in (0, π)

(ii) strictly increasing in (π, 2π)

(iii) neither increasing nor decreasing in (0, 2π)Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39(i)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(i)

Question 39(ii)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(ii)

Question 39(iii)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(iii)

Question 1(x)

Find the intervals in which the following functions are increasing or decreasing:

Solution 1(x)

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

Clearly, f'(x) > 0 if x < -2 or x > -1

And, f'(x) < 0 if -2 < x < -1

Thus, f(x) increases on  and decreases on Question 1(xxix)

Find the intervals in which the following functions are increasing or decreasing:

Solution 1(xxix)

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

The points x = 2, 4 and -3 divide the number line into four disjoint intervals namely 

Consider the interval 

In this case, x – 2 < 0, x – 4 < 0 and x + 3 < 0

Therefore, f'(x) < 0 when 

Thus the function is decreasing in 

Consider the interval 

In this case, x – 2 < 0, x – 4 < 0 and x + 3 > 0

Therefore, f'(x) > 0 when 

Thus the function is increasing in 

Now, consider the interval 

In this case, x – 2 > 0, x – 4 < 0 and x + 3 > 0

Therefore, f'(x) < 0 when 

Thus the function is decreasing in 

And now, consider the interval 

In this case, x – 2 > 0, x – 4 > 0 and x + 3 > 0

Therefore, f'(x) < 0 when 

Thus the function is increasing in Question 30(ii)

Prove that the following function is increasing on R:

Solution 30(ii)

Given: 

Differentiating w.r.t x, we get

Now, 

Hence, f(x) is an increasing function for all x.