Increasing & Decreasing Function Questions practice PDF

Increasing and decreasing functions — the study of monotonicity — is one of the most important applications of derivatives in Class 12 calculus, and it appears in almost every CBSE board paper. This page gives you 56 practice questions on the topic, transcribed from NCERT and CBSE exercises. They cover finding the intervals where a function increases or decreases, and proving that a given function is increasing or decreasing on a stated interval. Attempt them on paper, and use the button to print the set or save it as a PDF for revision or a class test.

What this worksheet covers

• Interval problems — polynomials such as \(f(x) = 2x^3 - 15x^2 + 36x + 1\), and functions like \(f(x) = (x+2)e^{-x}\).
• Proof problems — showing a function is increasing or decreasing, e.g. \(f(x) = x - \sin x\) or \(f(x) = \tan^{-1}x - x\).
• Trigonometric and logarithmic functions — such as \(\log \cos x\) and \(\sin x - \cos x\).
• Parameter questions — finding values of \(a\), \(b\), or \(k\) that make a function monotonic on \(\mathbb{R}\).

The key idea: the first derivative test

Almost every question here is solved using the sign of the first derivative \(f'(x)\):

• If \(f'(x) > 0\) on an interval, then \(f\) is increasing there.
• If \(f'(x) < 0\) on an interval, then \(f\) is decreasing there.
• The points where \(f'(x) = 0\) are the critical points that split the number line into intervals to test.

Method: find \(f'(x)\), solve \(f'(x) = 0\) to get the critical points, then check the sign of \(f'(x)\) in each resulting interval to decide where the function increases or decreases.

Who is this practice set for?

These questions are ideal for Class 12 / CBSE and NCERT students preparing for board exams, as well as anyone revising applications of derivatives. Teachers can use the printable PDF as a ready-made homework sheet, worksheet, or quick class test.

How to use this worksheet

Scroll to the question set below and work through the problems on paper. When you're ready to study offline, click Print / Save as PDF — it exports a clean copy of all 56 questions, ready to print or download.

Tags: #IncreasingDecreasingFunctions #Monotonicity #ApplicationsOfDerivatives #Calculus #Class12Maths #CBSE #NCERT #MathPractice #BoardExam #Derivatives #FirstDerivativeTest #Maths #MathWorksheet #PrintableMath #ExamPrep #StudyMaths #FreePDF

Class 12 · CBSE / NCERT · 56 Questions

Increasing & Decreasing Functions

Fifty-six questions on monotonicity (Exercise 13.2) — finding intervals where a function increases or decreases, and proving functions increasing or decreasing on given domains.

1

Polynomial

Find the intervals in which \(f(x) = 10 - 6x - 2x^2\) is increasing or decreasing.

2

Polynomial

Find the intervals in which \(f(x) = x^2 + 2x - 5\) is increasing or decreasing.

3

Polynomial

Find the intervals in which \(f(x) = 6 - 9x - x^2\) is increasing or decreasing.

4

Polynomial

Find the intervals in which \(f(x) = 2x^3 - 12x^2 + 18x + 15\) is increasing or decreasing.

5

Polynomial

Find the intervals in which \(f(x) = 5 + 36x + 3x^2 - 2x^3\) is increasing or decreasing.

6

Polynomial

Find the intervals in which \(f(x) = 5x^3 - 15x^2 - 120x + 3\) is increasing or decreasing.

7

Polynomial

Find the intervals in which \(f(x) = x^3 - 6x^2 - 36x + 2\) is increasing or decreasing.

8

Polynomial

Find the intervals in which \(f(x) = 2x^3 - 15x^2 + 36x + 1\) is increasing or decreasing.

9

Polynomial

Find the intervals in which \(f(x) = 2x^3 + 9x^2 + 12x + 20\) is increasing or decreasing.

10

Polynomial

Find the intervals in which \(f(x) = 2x^3 - 9x^2 + 12x - 5\) is increasing or decreasing.

11

Polynomial

Find the intervals in which \(f(x) = 2x^3 - 24x + 107\) is increasing or decreasing.

12

Polynomial

Find the intervals in which \(f(x) = -2x^3 - 9x^2 - 12x + 1\) is increasing or decreasing.

13

Polynomial

Find the intervals in which \(f(x) = (x - 1)(x - 2)^2\) is increasing or decreasing.

14

Polynomial

Find the intervals in which \(f(x) = x^3 - 12x^2 + 36x + 17\) is increasing or decreasing.

15

Polynomial

Find the intervals in which \(f(x) = 2x^3 - 24x + 7\) is increasing or decreasing.

16

Polynomial

Find the intervals in which \(f(x) = x^4 - 4x\) is increasing or decreasing.

17

Polynomial

Find the intervals in which \(f(x) = x^4 - 4x^3 + 4x^2 + 15\) is increasing or decreasing.

18

Fractional powers

Find the intervals in which \(f(x) = 5x^{3/2} - 3x^{5/2},\; x > 0\) is increasing or decreasing.

19

Polynomial

Find the intervals in which \(f(x) = x^8 + 6x^2\) is increasing or decreasing.

20

Polynomial

Find the intervals in which \(f(x) = x^3 - 6x^2 + 9x + 15\) is increasing or decreasing.

21

Polynomial

Find the intervals in which \(f(x) = \{x(x - 2)\}^2\) is increasing or decreasing.

22

Polynomial

Find the intervals in which \(f(x) = 3x^4 - 4x^3 - 12x^2 + 5\) is increasing or decreasing.

23

Polynomial

Find the intervals in which \(f(x) = \dfrac{3}{2}x^4 - 4x^3 - 45x^2 + 51\) is increasing or decreasing.

24

Polynomial

Find the intervals in which \(f(x) = \dfrac{x^4}{4} - x^3 - 5x^2 + 24x + 12\) is increasing or decreasing.

25

Logarithmic

Find the intervals in which \(f(x) = \log(2 + x) - \dfrac{2x}{2 + x},\; x \in \mathbb{R}\) is increasing or decreasing.

26

Trigonometric

Find the intervals in which \(f(x) = \sin x - \cos x,\; 0 < x < 2\pi\) is increasing or decreasing.

27

Logarithmic

Find the intervals in which \(f(x) = \log(1 + x) - \dfrac{x}{1 + x}\) is increasing or decreasing.

28

Exponential

Find the intervals in which \(f(x) = (x + 2)\,e^{-x}\) is increasing or decreasing.

29

Prove monotonic

Show that \(f(x) = e^{2x}\) is increasing on \(\mathbb{R}\).

30

Prove monotonic

Show that \(f(x) = e^{1/x},\; x \neq 0\) is a decreasing function for all \(x \neq 0\).

31

Prove monotonic

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

32

Prove monotonic

Show that \(f(x) = x - \sin x\) is increasing for all \(x \in \mathbb{R}\).

33

Prove monotonic

Show that \(f(x) = x^3 - 15x^2 + 75x - 50\) is an increasing function for all \(x \in \mathbb{R}\).

34

Prove monotonic

Show that \(f(x) = \cos^2 x\) is a decreasing function on \(\left(0, \tfrac{\pi}{2}\right)\).

35

Prove monotonic

Show that \(f(x) = \tan x\) is an increasing function on \(\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)\).

36

Prove monotonic

Show that \(f(x) = (x - 1)e^{x} + 1\) is an increasing function for all \(x > 0\).

37

Prove monotonic

Show that \(f(x) = \sin x - \cos x\) is an increasing function on \(\left(-\tfrac{\pi}{4}, \tfrac{\pi}{4}\right)\).

38

Prove monotonic

Show that \(f(x) = \tan^{-1}x - x\) is a decreasing function on \(\mathbb{R}\).

39

Prove monotonic

Determine whether \(f(x) = -\dfrac{x}{2} + \sin x\) is increasing or decreasing on \(\left(-\tfrac{\pi}{3}, \tfrac{\pi}{3}\right)\).

40

Prove monotonic

Show that \(f(x) = 3x^5 + 40x^3 + 240x\) is increasing on \(\mathbb{R}\).

41

Prove monotonic

Show that \(f(x) = 4x^3 - 18x^2 + 27x - 27\) is increasing on \(\mathbb{R}\).

42

Prove monotonic

Prove that \(f(x) = \log \cos x\) is strictly decreasing on \(\left(0, \tfrac{\pi}{2}\right)\).

43

Prove monotonic

Show that \(f(x) = \tan^{-1}(\sin x + \cos x)\) is a decreasing function on \(\left(\tfrac{\pi}{4}, \tfrac{\pi}{2}\right)\).

44

Prove monotonic

Show that \(f(x) = \cot^{-1}(\sin x + \cos x)\) is decreasing on \(\left(0, \tfrac{\pi}{4}\right)\) and increasing on \(\left(\tfrac{\pi}{4}, \tfrac{\pi}{2}\right)\).

45

Prove monotonic

Show that \(f(x) = x^9 + 4x^7 + 11\) is an increasing function for all \(x \in \mathbb{R}\).

46

Prove monotonic

Show that \(f(x) = \dfrac{16 \sin x}{4 + \cos x} - x\) is strictly decreasing in \(\left(\tfrac{\pi}{2}, \pi\right)\).

47

Prove monotonic

Show that \(f(x) = x^2 - x \sin x\) is an increasing function on \(\left(0, \tfrac{\pi}{2}\right)\).

48

Find the parameter

Find the value(s) of \(a\) for which \(f(x) = x^3 - ax\) is an increasing function on \(\mathbb{R}\).

49

Find the parameter

Find the values of \(b\) for which \(f(x) = \sin x - bx + c\) is a decreasing function on \(\mathbb{R}\).

50

Short answer

Write the set of values of \(x\) for which \(f(x) = \tan x - x\) is increasing.

51

Short answer

If \(k\pi\) is the length of the largest interval in which \(f(x) = 3\sin x - 4\sin^3 x\) is increasing, find \(k\).

52

Short answer

Write the largest interval in which \(f(x) = x^{1/x}\) is strictly increasing.

53

Short answer

What are the values of \(a\) for which \(f(x) = a^{x}\) is increasing on \(\mathbb{R}\)?

54

Short answer

Write the set of values of \(k\) for which \(f(x) = kx - \sin x\) is increasing on \(\mathbb{R}\).

55

Short answer

Write the set of values of \(a\) for which \(f(x) = \cos x + a^2 x + b\) is strictly increasing on \(\mathbb{R}\).

56

Short answer

If \(f(x) = a(\tan x - \cot x)\), where \(a > 0\), find whether \(f(x)\) is increasing or decreasing in its domain.