Solution of Chapter 17. Increasing and Decreasing Functions (RD Sharma - Mathematics (Volume 1) Book)

Exercise 17.1

1

Prove that the function f(x) = loge x is increasing on (0, ∞).

2

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decresing on (0, ∞), if 0 < a < 1.

3

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.

4

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.

5

Show that is a decreasing function on (0, ∞).

6

Show that decreases in the interval [0, ∞) and increases in the interval (-∞, 0].

7

Show that is neither increasing nor decreasing on R.

8

Without using the derivative, show that the function f(x) = | x | is

A. strictly increasing in (0, ∞)

B. strictly decreasing in (-∞, 0).

9

Without using the derivative show that the function f(x) = 7x - 3 is strictly increasing function on R.

Exercise 17.2

1

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

f(x) = 10 – 6x – 2x2

1

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

f(x) = x2 + 2x – 5

1

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

f(x) = 6 – 9x – x2

1

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

f(x) = 2x3 – 12x2 + 18x + 15

1

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

f(x) = 5 + 36x + 3x2 – 2x3

1

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

f(x) = 8 + 36x + 3x2 – 2x3

1

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

f(x) = 5x3 – 15x2 – 120x + 3

1

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

f(x) = x3 – 6x2 – 36x + 2

1

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

f(x) = 2x3 – 15x2 + 36x + 1

1

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

f(x) = 2x3 + 9x2 + 12x + 20

1

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

f(x) = 2x3 – 9x2 + 12x – 5

1

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

f(x) = 6 + 12x + 3x2 – 2x3

1

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

f(x) = 2x3 – 24x + 107

1

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

f(x) = – 2x3 – 9x2 – 12x + 1

1

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

f(x) = (x – 1) (x – 2)2

1

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

f(x) = x3 – 12x2 + 36x + 17

1

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

f(x) = 2x3 – 24x + 7

1

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

1

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

f(x) = x4 – 4x

1

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

1

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

f(x) = x4 – 4x3 + 4x2 + 15

1

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

1

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

f(x) = x8 + 6x2

1

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

f(x) = x3 – 6x2 + 9x + 15

1

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

f(x) = {x (x – 2)}2

1

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

f(x) = 3x4 – 4x3 – 12x2 + 5

1

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

1

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

2

Determine the values of x for which the function f(x) = x2 – 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 – 6x + 9 where the normal is parallel to the line
y = x + 5.

3

Find the intervals in which f(x) = sin x – cos x, where 0 < x < 2π is increasing or decreasing.

4

Show that f(x) = e2x is increasing on R.

5

Show that , x ≠ 0 is a decreasing function for all x ≠ 0.

6

Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0.

7

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π).

8

Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).

9

Show that f(x) = x – sin x is increasing for all x ϵ R.

10

Show that f(x) = x3 – 15x2 + 75x – 50 is an increasing function for all x ϵ R.

11

Show that f(x) = cos2 x is a decreasing function on (0, π/2).

12

Show that f(x) = sin x is an increasing function on (–π/2, π/2).

13

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (–π, 0) and neither increasing nor decreasing in (–π, π).

14

Show that f(x) = tan x is an increasing function on (–π/2, π/2).

15

Show that f(x) = tan–1 (sin x + cos x) is a decreasing function on the interval (π/4, π /2).

16

Show that the function is decreasing on

17

Show that the function f(x) = cot–1 (sin x + cos x) is decreasing on (0, π/4) and increasing on (π/4, π/2).

18

Show that f(x) = (x – 1) ex + 1 is an increasing function for all x > 0.

19

Show that the function x2 – x + 1 is neither increasing nor decreasing on (0, 1).

20

Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ϵ R.

21

Prove that the function f(x) = x3 – 6x2 + 12x – 18 is increasing on R.

22

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 – 6x + 3 is increasing on the interval [4, 6].

23

Show that f(x) = sin x – cos x is an increasing function on (–π /4, π /4)?

24

Show that f(x) = tan–1 x – x is a decreasing function on R ?

25

Determine whether f(x) = x/2 + sin x is increasing or decreasing on (–π /3, π/3) ?

26

Find the interval in which is increasing or decreasing ?

27

Find the intervals in which f(x) = (x + 2)e–x is increasing or decreasing ?

28

Show that the function f given by f(x) = 10x is increasing for all x ?

29

Prove that the function f given by f(x) = x – [x] is increasing in (0, 1) ?

30

Prove that the following function is increasing on r?

i. f(x) = 3x5 + 40x3 + 240x

ii. f(x) = 4x3 – 18x2 + 27x – 27

31

Prove that the function f given by f(x) = log cos x is strictly increasing on (–π/2, 0) and strictly decreasing on (0, π/2) ?

32

Prove that the function f given by f(x) = x3 – 3x2 + 4x is strictly increasing on R ?

33

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

i. strictly decreasing on (0, π)

ii. strictly increasing in (π, 2π)

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

34

Show that f(x) = – x sin x is an increasing function on (0, π/2) ?

35

Find the value(s) of a for which f(x) = – ax is an increasing function on R ?

36

Find the values of b for which the function f(x) = sin x – bx + c is a decreasing function on R ?

37

Show that f(x) = x + cos x – a is an increasing function on R for all values of a ?

38

Let F defined on [0, 1] be twice differentiable such that | f”(x) ≤ 1 for all x ϵ [0, 1]. If f(0) = f(1), then show that |f’(x) | < 1 for all x ϵ [0, 1] ?

39

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

i. f(x) = x |x|, x ϵ R

ii. f(x) = sin x + |sin x|, 0 < x ≤ 2 π

iii. f(x) = sin x (1 + cos x), 0 < x < π/2

MCQ

1

Mark the correct alternative in the following:

The interval of increase of the function f(x) = x – ex + tan is

2

Mark the correct alternative in the following:

The function f(x) = cos–1 x + x increases in the interval.

3

Mark the correct alternative in the following:

The function f(x) = xx decreases on the interval.

4

Mark the correct alternative in the following:

The function f(x) = 2log(x – 2) – x2 + 4x + 1 increases on the interval.

5

Mark the correct alternative in the following:

If the function f(x) = 2x2 – kx + 5 is increasing on [1, 2], then k lies in the interval.

6

Mark the correct alternative in the following:

Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.

7

Mark the correct alternative in the following:

The function is of the following types:

8

Mark the correct alternative in the following:

If the function f(x) = 2tanx + (2a + 1) loge |sec x| + (a – 2) x is increasing on R, then

9

Mark the correct alternative in the following:

Let f(x) = tan–1 (g(x)), where g(x) is monotonically increasing for Then, f(x) is

10

Mark the correct alternative in the following:

Let f(x) = x3 – 6x2 + 15x + 3. Then,

11

Mark the correct alternative in the following:

The function f(x) = x2 e-x is monotonic increasing when

12

Mark the correct alternative in the following:

Function f(x) = cosx – 2λ x is monotonic decreasing when

13

Mark the correct alternative in the following:

In the interval (1, 2), function f(x) = 2 |x – 1|+3|x – 2| is

14

Mark the correct alternative in the following:

Function f(x) = x3– 27x +5 is monotonically increasing when

15

Mark the correct alternative in the following:

Function f(x) = 2x3 – 9x2 + 12x + 29 is monotonically decreasing when

16

Mark the correct alternative in the following:

If the function f(x) = kx3 – 9x2 + 9x + 3 is monotonically increasing in every interval, then

17

Mark the correct alternative in the following:

f(x) = 2x – tan–1 x – logis monotonically increasing when

18

Mark the correct alternative in the following:

Function f(x) = |x| – |x – 1| is monotonically increasing when

19

Mark the correct alternative in the following:

Every invertible function is

20

Mark the correct alternative in the following:

In the interval (1, 2), function f(x) = 2|x – 1|+3 |x – 2| is

21

Mark the correct alternative in the following:

If the function f(x) = cos|x| – 2ax + b increases along the entire number scale, then

22

Mark the correct alternative in the following:

The function is

23

Mark the correct alternative in the following:

The function is increasing, if

24

Mark the correct alternative in the following:

Function f(x) = ax is increasing or R, if

25

Mark the correct alternative in the following:

Function f(x) = loga x is increasing on R, if

26

Mark the correct alternative in the following:

Let ϕ(x) = f(x) + f(2a – x) and f’’(x) > 0 for all xϵ[0, a]. The, ϕ(x)

27

Mark the correct alternative in the following:

If the function f(x) = x2 – kx + 5 is increasing on [2, 4], then

28

Mark the correct alternative in the following:

The function defined on is

29

Mark the correct alternative in the following:

If the function f(x) = x3 – 9k x2 + 27x + 30 is increasing on R, then