##### VMC VIQ Scholarship Test

Register for Vidyamandir Intellect Quest. Get Scholarship and Cash Rewards.

Edited By Ramraj Saini | Updated on Dec 03, 2023 08:40 PM IST | #CBSE Class 12th

**NCERT Solutions for Exercise 6.2 Class 12 Maths Chapter 6 Application of Derivatives **are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the **latest syllabus and pattern of CBSE 2023-24. **NCERT solutions for exercise 6.2 Class 12 Maths chapter 6 gives an insight into topic 6.3 increasing and decreasing functions. Before exercise 6.2 Class 12 Maths, NCERT has explained the questions and examples related to the rate of change of quantities. After the NCERT solutions for Class 12 Maths chapter 6 exercise 6.2 the concepts of decreasing and increasing functions is introduced in the NCERT book and then certain theorems are discussed followed by example questions and Class 12th Maths chapter 6 exercise 6.2.

The NCERT solutions for Class 12 Maths chapter 6 exercise 6.2 gives practice on topic 6.3 of Class 12 Maths NCERT syllabus. Solving the Class 12 Maths chapter 6 exercise 6.2 gives more knowledge of the concepts of increasing and decreasing functions. The following exercises are also discussed in the chapter application of derivatives. **12th class Maths exercise 6.2 **answers** **are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

- Application of Derivatives Exercise 6.1
- Application of Derivatives Exercise 6.3
- Application of Derivatives Exercise 6.4
- Application of Derivatives Exercise 6.5
- Application of Derivatives Miscellaneous Exercise

** Question:1 ** . Show that the function given by f (x) = 3x + 17 is increasing on R.

** Answer: **

Let are two numbers in R

Hence, f is strictly increasing on R

** Question:2. ** Show that the function given by is increasing on R.

** Answer: **

Let are two numbers in R

Hence, the function is strictly increasing in R

** Question:3 a) ** Show that the function given by f (x) = is increasing in

** Answer: **

Given f(x) = sinx

Since,

Hence, f(x) = sinx is strictly increasing in

** Question:3 ** ** b) ** Show that the function given by f (x) = is

** Answer: **

f(x) = sin x

Since, for each

So, we have

Hence, f(x) = sin x is strictly decreasing in

** Question:3 ** ** c) ** Show that the function given by f (x) = is neither increasing nor decreasing in

** Answer: **

We know that sin x is strictly increasing in and strictly decreasing in

So, by this, we can say that f(x) = sinx is neither increasing or decreasing in range

** Question:4(a). ** Find the intervals in which the function f given by is increasing

** Answer: **

Now,

4x - 3 = 0

So, the range is

So,

when Hence, f(x) is strictly decreasing in this range

and

when Hence, f(x) is strictly increasing in this range

Hence, is strictly increasing in

** Question:4(b) ** Find the intervals in which the function f given by is

decreasing

** Answer: **

Now,

4x - 3 = 0

So, the range is

So,

when Hence, f(x) is strictly decreasing in this range

and

when Hence, f(x) is strictly increasing in this range

Hence, is strictly decreasing in

** Question:5(a) ** Find the intervals in which the function f given by is

increasing

** Answer: **

It is given that

So,

x = -2 , x = 3

So, three ranges are there

Function is positive in interval and negative in the interval (-2,3)

Hence, is strictly increasing in

and strictly decreasing in the interval (-2,3)

** Question:5(b) ** Find the intervals in which the function f given by is

decreasing

** Answer: **

We have _{}

Differentiating the function with respect to x, we get :

or

When , we have :

or

So, three ranges are there

Function is positive in the interval and negative in the interval (-2,3)

So, f(x) is decreasing in (-2, 3)

** Question:6(a) ** Find the intervals in which the following functions are strictly increasing or

decreasing:

** Answer: **

f(x) =

Now,

The range is from

In interval is -ve

Hence, function f(x) = is strictly decreasing in interval

In interval is +ve

Hence, function f(x) = is strictly increasing in interval

** Question:6(b) ** Find the intervals in which the following functions are strictly increasing or

decreasing

** Answer: **

Given function is,

Now,

So, the range is

In interval , is +ve

Hence, is strictly increasing in the interval

In interval , is -ve

Hence, is strictly decreasing in interval

** Question:6(c) ** Find the intervals in which the following functions are strictly increasing or

decreasing:

** Answer: **

Given function is,

Now,

So, the range is

In interval , is -ve

Hence, is strictly decreasing in interval

In interval (-2,-1) , is +ve

Hence, is strictly increasing in the interval (-2,-1)

** Question:6(d) ** Find the intervals in which the following functions are strictly increasing or

decreasing:

** Answer: **

Given function is,

Now,

So, the range is

In interval , is +ve

Hence, is strictly increasing in interval

In interval , is -ve

Hence, is strictly decreasing in interval

** Question:6(e) ** Find the intervals in which the following functions are strictly increasing or

decreasing:

** Answer: **

Given function is,

Now,

So, the intervals are

Our function is +ve in the interval

Hence, is strictly increasing in the interval

Our function is -ve in the interval

Hence, is strictly decreasing in interval

** Question:7 ** Show that is an increasing function of x throughout its domain.

** Answer: **

Given function is,

Now, for , is is clear that

Hence, strictly increasing when

** Question:8 ** Find the values of x for which is an increasing function.

** Answer: **

Given function is,

Now,

So, the intervals are

In interval ,

Hence, is an increasing function in the interval

** Question:9 ** Prove that is an increasing function of

** Answer: **

Given function is,

Now, for

So,

Hence, is increasing function in

** Question:10 ** Prove that the logarithmic function is increasing on

** Answer: **

Let logarithmic function is log x

Now, for all values of x in ,

Hence, the logarithmic function is increasing in the interval

** Question:11 ** Prove that the function f given by is neither strictly increasing nor decreasing on (– 1, 1).

** Answer: **

Given function is,

Now, for interval , and for interval

Hence, by this, we can say that is neither strictly increasing nor decreasing in the interval (-1,1)

** Question:12 ** Which of the following functions are decreasing on

** Answer: **

(A)

for x in

Hence, is decreasing function in

(B)

Now, as

for 2x in

Hence, is decreasing function in

(C)

Now, as

for and

Hence, it is clear that is neither increasing nor decreasing in

(D)

for x in

Hence, is strictly increasing function in the interval

So, only (A) and (B) are decreasing functions in

** Answer: **

(A) Given function is,

Now, in interval (0,1)

Hence, is increasing function in interval (0,1)

(B) Now, in interval

,

Hence, is increasing function in interval

(C) Now, in interval

,

Hence, is increasing function in interval

So, is increasing for all cases

Hence, correct answer is (D) None of these

** Question:14 ** For what values of a the function f given by is increasing on

[1, 2]?

** Answer: **

Given function is,

Now, we can clearly see that for every value of

Hence, is increasing for every value of in the interval [1,2]

** Question:15 ** Let I be any interval disjoint from [–1, 1]. Prove that the function f given by is increasing on I.

** Answer: **

Given function is,

Now,

So, intervals are from

In interval ,

Hence, is increasing in interval

In interval (-1,1) ,

Hence, is decreasing in interval (-1,1)

Hence, the function f given by is increasing on I disjoint from [–1, 1]

** Question:16 ** Prove that the function f given by is increasing on

Given function is,

Now, we know that cot x is+ve in the interval and -ve in the interval

Hence, is increasing in the interval and decreasing in interval

** Question:17 ** Prove that the function f given by f (x) = log |cos x| is decreasing on

and increasing on

** Answer: **

Given function is,

f(x) = log|cos x|

value of cos x is always +ve in both these cases

So, we can write log|cos x| = log(cos x)

Now,

We know that in interval ,

Hence, f(x) = log|cos x| is decreasing in interval

We know that in interval ,

Hence, f(x) = log|cos x| is increasing in interval

** Question:18 ** Prove that the function given by is increasing in R.

** Answer: **

Given function is,

We can clearly see that for any value of x in R

Hence, is an increasing function in R

** Question:19 ** The interval in which is increasing is

(A) (B) (C) (D)

** Answer: **

Given function is,

Now, it is clear that only in the interval (0,2)

So, is an increasing function for the interval (0,2)

Hence, (D) is the answer

The questions discussed in the Class 12th Maths chapter 6 exercise 6.2 uses differentiation to find out the increasing and decreasing function. The NCERT Class 12 Maths Book explains the increasing and decreasing functions with suitable examples and graphical representations. All the examples in the NCERT Book and the NCERT solutions for Class 12 Maths chapter 6 exercise 6.2 are important from the exam point of view.

**Also Read| **Application of Derivatives Class 12 Notes

Exercise 6.2 Class 12 Maths helps students to grasp the concepts in a better way.

NCERT solutions for Class 12 Maths chapter 6 exercise 6.2 is useful for the preparation of board exams that follows the NCERT Syllabus

Along with this students can also refer to the NCERT exemplar solutions of the same chapter for a good score.

JEE Main Highest Scoring Chapters & Topics

Just Study 40% Syllabus and Score upto 100%

Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in ex 6.2 class 12, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 maths ex 6.2, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 ex 6.2 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this 12th class maths exercise 6.2 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for ex 6.2 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for class 12 maths ex 6.2 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

**Also see-**

- NCERT Solutions Class 12 Chemistry
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Biology
- NCERT Solutions for Class 12 Mathematics

1. The number of examples given in the Class 12 NCERT Maths topic 6.3 is………...

7 examples are discussed in the NCERT topic decreasing and increasing function.

2. What is the topic discussed before increasing and decreasing function?

The topic 6.2 rates of change of quantities are discussed prior to increasing and decreasing function

3. Why do we study applications of derivatives?

The concepts of derivatives and their applications are used in various engineering and science domains for analysis purposes. So to build the basics of derivatives the NCERT Mathematics Books Classes 11 and 12 introduce the concepts of derivatives.

4. Name the topics discussed after NCERT Solutions for Class 12 Maths chapter 6 exercise 6.2?

Tangents and normal is the topic discussed after the exercise 6.2 Class 12 Maths

5. In the interval (0, pi/2) the function sinx is …………….

The function sinx is strictly increasing in the open interval (0, pi/2)

6. Whether f(x)=sinx is increasing or decreasing in (pi/2, pi)

If we look at the graph of sinx it can be seen that f(x) = sinx is strictly decreasing.

7. What is the nature of sinx in open interval (0, pi)?

Sinx is neither increasing nor decreasing in the given interval (0, pi)

8. Name the chapter coming after the application of derivatives in the NCERT book for Class 12 Maths

Integrals is introduced in chapter 7 of Class 12 NCERT Maths.

Application Date:05 September,2024 - 20 September,2024

Admit Card Date:13 September,2024 - 07 October,2024

Admit Card Date:13 September,2024 - 07 October,2024

Application Date:17 September,2024 - 30 September,2024

Full-Stack Web Development

Via Masai School Artificial Intelligence Projects

Via Great Learning Digital Banking Business Model

Via State Bank of India Introduction to Watson AI

Via IBM Business Analytics Foundations

Via PW Skills Edx

1113 coursesCoursera

804 coursesUdemy

394 coursesFuturelearn

222 coursesHarvard University, Cambridge

98 coursesIBM

85 coursesUniversity of Essex, Colchester

Wivenhoe Park Colchester CO4 3SQUniversity College London, London

Gower Street, London, WC1E 6BTThe University of Edinburgh, Edinburgh

Old College, South Bridge, Edinburgh, Post Code EH8 9YLUniversity of Bristol, Bristol

Beacon House, Queens Road, Bristol, BS8 1QUUniversity of Delaware, Newark

Newark, DE 19716University of Nottingham, Nottingham

University Park, Nottingham NG7 2RDETH Zurich Scholarship 2024: Requirements and Deadlines

3 minIELTS Exam 2024 - Dates, Registration, Syllabus, Fees, Pattern, Result

11 minTOEFL Syllabus 2024 - Listening, Reading, Speaking and Writing Section

9 minGMAT Exam 2024: Dates, Fees, Registration, Syllabus, Pattern, Result

17 minEngineering in USA for Indian Students: Top Universities, Fees and Scholarships

6 minGMAT Test Centres 2024: List of City Wise Exam Centres in India

6 minHave a question related to CBSE Class 12th ?

You can use them people also used problem

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

To apply, download the Medhavi App from the Google Play Store, sign up, and read the detailed notification about the scholarship exam. Complete the registration within the app, take the exam from home using the app, and receive your results within two days. Following this, upload the necessary documents and bank account details for verification. Upon successful verification, the scholarship amount will be directly transferred to your bank account.

The scholarships are categorized based on the marks obtained in the exam: Type A for those scoring 60% or above, Type B for scores between 50% and 60%, and Type C for scores between 40% and 50%. The cash scholarships range from Rs. 2,000 to Rs. 18,000 per month, depending on the exam and the marks obtained.

Since you already have a 12th-grade qualification with 84%, you meet the eligibility criteria and can apply for the Medhavi Scholarship exam. Preparing well for the exam can increase your chances of receiving a higher scholarship.

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

**
Thank you and wishing you all the best for your bright future.
**

Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

**
Good luck with your studies!
**

Register for Vidyamandir Intellect Quest. Get Scholarship and Cash Rewards.

Register for Tallentex '25 - One of The Biggest Talent Encouragement Exam

As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters

As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters

Accepted by more than 11,000 universities in over 150 countries worldwide

Register now for PTE & Unlock 10% OFF : Use promo code: 'C360SPL10'. Limited Period Offer! Trusted by 3,500+ universities globally

News and Notifications

September 12, 2024 - 01:16 PM IST

September 02, 2024 - 12:13 PM IST

August 28, 2024 - 04:58 PM IST

August 21, 2024 - 10:24 PM IST

August 06, 2024 - 12:37 PM IST

August 02, 2024 - 04:46 PM IST

July 30, 2024 - 04:48 PM IST

Back to top