NCERT Solutions for Exercise 6.2 Class 12 Maths Chapter 10 - Application of Derivatives
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.
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
Application of Derivatives Class 12 Exercise 6.2
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
decreasing in
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
Question:13 On which of the following intervals is the function f given by decreasing ?
(A) (0,1) (B) (C) (D) None of these
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
Answer:
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
More About NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.2
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
Benefits of NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.2
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.
Also see-
NCERT Exemplar Solutions Class 12 Maths Chapter 6
NCERT Solutions for Class 12 Maths Chapter 6
NCERT Solutions Subject Wise
NCERT Solutions Class 12 Chemistry
NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Biology
NCERT Solutions for Class 12 Mathematics
Subject Wise NCERT Exemplar Solutions
NCERT Exemplar Class 12 Maths
NCERT Exemplar Class 12 Physics
NCERT Exemplar Class 12 Chemistry
NCERT Exemplar Class 12 Biology