NEET/JEE Coaching Scholarship
Get up to 90% Scholarship on Offline NEET/JEE coaching from top Institutes
Where the derivative is zero, the world holds its breath — will it rise, fall, or simply turn? Imagine you are running a 100-metre race. Your speed at each moment is the first derivative of your position, which tells how your position is changing. The second derivative, which shows how quickly your speed changes, is the acceleration. In the NCERT Solutions for Exercise 6.2 Class 12 Maths Chapter 6 Application of Derivatives, students will gain the knowledge of uses of derivatives to solve real-life problems like finding the highest or lowest value of something, checking if a function is increasing or decreasing, and analysing how things change. In this specific section of the NCERT, students will get familiar with the behaviour of the function: if it is going up, then it is increasing, or if it is going down, then it is decreasing.
As per the reports, the Central Board of Secondary Education (CBSE) will declare the Class 10, 12 board exams 2025 in the mid-May on its official website at cbse.gov.in and results.cbse.nic.in. Notably, students will also be able to check their marks through Digilocker with the help of access codes.
Subject matter experts with multiple years of experience have curated these NCERT solutions to support students in mastering these key concepts.
Question 1: . Show that the function given by f (x) = 3x + 17 is increasing on R.
Answer:
Let
Hence, f is strictly increasing on R
Question 2: Show that the function given by
Answer:
Hence, the function
Question 3: (a) Show that the function given by f (x) =
Answer:
Given f(x) = sinx
Hence, f(x) = sinx is strictly increasing in
Question 3: (b) Show that the function given by f (x) =
Answer:
f(x) = sin x
Since,
So, we have
Hence, f(x) = sin x is strictly decreasing in
Question 3: (c) Show that the function given by f (x) =
Answer:
We know that sin x is strictly increasing 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
Answer:
Now,
4x - 3 = 0
So,
and
Hence,
Question 4: (b) Find the intervals in which the function f given by
decreasing
Answer:
Now,
4x - 3 = 0
So,
and
Hence,
Question 5:(a) Find the intervals in which the function f given by
increasing
Answer:
It is given that
So,
x = -2 , x = 3
Function
Hence,
and strictly decreasing in the interval (-2,3)
Question 5:(b) Find the intervals in which the function f given by
decreasing
Answer:
We have
Differentiating the function with respect to x, we get :
or
When
or
Function
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,
In interval
Hence, function f(x) =
In interval
Hence, function f(x) =
Question 6:(b) Find the intervals in which the following functions are strictly increasing or
decreasing
Answer:
Given function is,
Now,
In interval
Hence,
In interval
Hence,
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
Hence,
In interval (-2,-1) ,
Hence,
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
Hence,
In interval
Hence,
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
Hence,
Our function
Hence,
Question 7: Show that
Answer:
Given function is,
Now, for
Hence,
Question 8: Find the values of x for which
Answer:
Given function is,
Now,
So, the intervals are
In interval
Hence,
Question 9: Prove that
Answer:
Given function is,
Now, for
So,
Hence,
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
Question 11: Prove that the function f given by
Answer:
Given function is,
Now, for interval
Hence, by this, we can say that
Question 12: Which of the following functions are decreasing on
Answer:
(A)
Hence,
(B)
Now, as
Hence,
(C)
Now, as
Hence, it is clear that
(D)
Hence,
So, only (A) and (B) are decreasing functions in
Question 13: On which of the following intervals is the function f given by
(A) (0,1)
Answer:
(A) Given function is,
Now, in interval (0,1)
Hence,
(B) Now, in interval
Hence,
(C) Now, in interval
Hence,
So,
Hence, correct answer is None of these
Question 14: For what values of a the function f given by
[1, 2]?
Answer:
Given function is,
Now, we can clearly see that for every value of
Hence,
Question 15: Let I be any interval disjoint from [–1, 1]. Prove that the function f given by
Answer:
Given function is,
Now,
So, intervals are from
In interval
Hence,
In interval (-1,1) ,
Hence,
Hence, the function f given by
Question 16 : Prove that the function f given by
Answer:
Given function is,
Now, we know that
Hence,
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
Answer:
Given function is,
We can clearly see that for any value of x in R
Hence,
Question 19: The interval in which
(A)
(B)
(C)
(D)
Answer:
Given function is,
Now, it is clear that
So,
Hence,
Also, read,
Increasing and decreasing function: Let I be an interval contained in the domain of a real-valued function
(i) increasing on I if
(ii) decreasing on I, if
(iii) constant on I, if
(iv) strictly increasing on I if
(v) strictly decreasing on I if
Also, read,
Unlock chapter-by-chapter guidance of NCERT solutions of other subjects using the links below.
Tap in these links below for detailed NCERT exemplar solutions for other subjects.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
7 examples are discussed in the NCERT topic decreasing and increasing function.
The topic 6.2 rates of change of quantities are discussed prior to increasing and decreasing function
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.
Tangents and normal is the topic discussed after the exercise 6.2 Class 12 Maths
The function sinx is strictly increasing in the open interval (0, pi/2)
If we look at the graph of sinx it can be seen that f(x) = sinx is strictly decreasing.
Sinx is neither increasing nor decreasing in the given interval (0, pi)
Integrals is introduced in chapter 7 of Class 12 NCERT Maths.
Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
Possible steps:
Re-evaluate Your Study Strategies:
Consider Professional Help:
Explore Alternative Options:
Focus on NEET 2025 Preparation:
Seek Support:
Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.
I hope this information helps you.
Hi,
Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.
Scholarship Details:
Type A: For candidates scoring 60% or above in the exam.
Type B: For candidates scoring between 50% and 60%.
Type C: For candidates scoring between 40% and 50%.
Cash Scholarship:
Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).
Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.
Hope you find this useful!
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.
Register for ALLEN Scholarship Test & get up to 90% Scholarship
Get up to 90% Scholarship on Offline NEET/JEE coaching from top Institutes
This ebook serves as a valuable study guide for NEET 2025 exam.
This e-book offers NEET PYQ and serves as an indispensable NEET study material.
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