# 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