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NCERT Solutions for Exercise 6.2 Class 12 Maths Chapter 10 - Application of Derivatives

NCERT Solutions for Exercise 6.2 Class 12 Maths Chapter 10 - Application of Derivatives

Edited By Komal Miglani | Updated on Apr 27, 2025 09:50 AM IST | #CBSE Class 12th

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.

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Subject matter experts with multiple years of experience have curated these NCERT solutions to support students in mastering these key concepts.

Class 12 Maths Chapter 6 Exercise 6.2 Solutions: Download PDF

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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 x1 and x2 are two numbers in R.

x1<x23x1<3x23x1+17<3x2+17f(x1)<f(x2)
Hence, f is strictly increasing on R

Question 2: Show that the function given by f(x)=e2x is increasing on R.

Answer:

Let x1 and x2 be two numbers in R.
x1 < x22x1<2x2e2x1<e2x2f(x1)<f(x2)
Hence, the function f(x)=e2x is strictly increasing in R

Question 3: (a) Show that the function given by f (x) = sinx is increasing in (0,π/2)

Answer:

Given f(x) = sinx
f(x)=cosx

Since, cosx>0 for each x(0,π2)

f(x)>0
Hence, f(x) = sinx is strictly increasing in (0,π2)

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

decreasing in (π2,π)

Answer:

f(x) = sin x
f(x)=cosx
Since, cosx<0 for each x ϵ(π2,π)
So, we have f(x)<0
Hence, f(x) = sin x is strictly decreasing in (π2,π)

Question 3: (c) Show that the function given by f (x) = sinx is neither increasing nor decreasing in (0,π)

Answer:

We know that sin x is strictly increasing in (0,π2) and strictly decreasing in (π2,π)
So, by this, we can say that f(x) = sinx is neither increasing or decreasing in range (0,π)

Question 4: (a). Find the intervals in which the function f given by f(x)=2x23x is increasing

Answer:

f(x)=2x23x
f(x)=4x3
Now,
f(x)=0
4x - 3 = 0
x=34
1628071298489

So, the range is (,34) and (34,)

So,
f(x)<0 when x ϵ(,34) Hence, f(x) is strictly decreasing in this range
and
f(x)>0 when xϵ(34,) Hence, f(x) is strictly increasing in this range
Hence, f(x)=2x23x is strictly increasing in xϵ(34,)

Question 4: (b) Find the intervals in which the function f given by f(x)=2x23x is
decreasing

Answer:

f(x)=2x23x
f(x)=4x3
Now,
f(x)=0
4x - 3 = 0
x=34
1654596806436

So, the range is (,34) and (34,)

So,
f(x)<0 when x ϵ(,34) Hence, f(x) is strictly decreasing in this range
and
f(x)>0 when xϵ(34,) Hence, f(x) is strictly increasing in this range
Hence, f(x)=2x23x is strictly decreasing in xϵ(,34)

Question 5:(a) Find the intervals in which the function f given by f(x)=2x33x236x+7 is
increasing

Answer:

It is given that
f(x)=2x33x236x+7
So,
f(x)=6x26x36
f(x)=0
6x26x36=06(x2x6)
x2x6=0
x23x+2x6=0
x(x3)+2(x3)=0
(x+2)(x3)=0
x = -2 , x = 3

So, three ranges are there (,2), (2,3) and (3,)

Function f(x)=6x26x36 is positive in interval (,2),(3,) and negative in the interval (-2,3)
Hence, f(x)=2x33x236x+7 is strictly increasing in (,2)(3,)
and strictly decreasing in the interval (-2,3)

Question 5:(b) Find the intervals in which the function f given by f(x)=2x33x236x+7 is
decreasing

Answer:

We have f(x)=2x33x236x+7

Differentiating the function with respect to x, we get :

f(x)=6x26x36

or =6(x3)(x+2)

When f(x) = 0 , we have :

0 =6(x3)(x+2)

or (x3)(x+2) = 0

1654597172033

So, three ranges are there (,2), (2,3), and (3,)
Function f(x)=6x26x36 is positive in the interval (,2),(3,) 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:
x2+2x5

Answer:

f(x) = x2+2x5
f(x)=2x+2=2(x+1)
Now,
f(x)=02(x+1)=0x=1

The range is from (,1)  and  (1,)

In interval (,1) f(x)=2(x+1) is -ve
Hence, function f(x) = x2+2x5 is strictly decreasing in interval (,1)
In interval (1,) f(x)=2(x+1) is +ve
Hence, function f(x) = x2+2x5 is strictly increasing in interval (1,)

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

106x2x2

Answer:

Given function is,
f(x)=106x2x2
f(x)=64x
Now,
f(x)=0
6+4x=0
x=32

So, the range is (,32)  and  (32,)
In interval (,32) , f(x)=64x is +ve
Hence, f(x)=106x2x2 is strictly increasing in the interval (,32)
In interval (32,) , f(x)=64x is -ve
Hence, f(x)=106x2x2 is strictly decreasing in interval (32,)

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

2x39x212x+1

Answer:

Given function is,
f(x)=2x39x212x+1
f(x)=6x218x12
Now,

f(x)=0

6x218x12=0

6(x2+3x+2)=0

x2+3x+2=0

x2+x+2x+2=0

x(x+1)+2(x+1)=0

(x+2)(x+1)=0

x=2 and x=1

So, the range is (,2), (2,1), and (1,).
In interval (,2) (1,) , f(x)=6x218x12 is -ve
Hence, f(x)=2x39x212x+1 is strictly decreasing in interval (,2) (1,)
In interval (-2,-1) , f(x)=6x218x12 is +ve
Hence, f(x)=2x39x212x+1 is strictly increasing in the interval (-2,-1)

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

69xx2

Answer:

Given function is,
f(x)=69xx2
f(x)=92x
Now,
f(x)=092x=02x=9x=92

So, the range is (,92) and (92,).
In interval (,92) , f(x)=92x is +ve
Hence, f(x)=69xx2 is strictly increasing in interval (,92)
In interval (92,) , f(x)=92x is -ve
Hence, f(x)=69xx2 is strictly decreasing in interval (92,)

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

(x+1)3(x3)3

Answer:

Given function is,
f(x)=(x+1)3(x3)3
f(x)=3(x+1)2(x3)3+3(x3)2(x+1)3
Now,

f(x)=0

3(x+1)2(x3)3+3(x3)2(x+1)3

3(x+1)2(x3)2((x3)+(x+1))=0

(x+1)(x3)=0 or (2x2)=0

x=1 and x=3 or x=1

So, the intervals are (,1), (1,1), (1,3) and (3,).
Our function f(x)=3(x+1)2(x3)3+3(x3)2(x+1)3 is +ve in the interval (1,3) and (3,)

Hence, f(x)=(x+1)3(x3)3 is strictly increasing in the interval (1,3) and (3,).

Our function f(x)=3(x+1)2(x3)3+3(x3)2(x+1)3 is negative in the interval (,1) and (1,1).

Hence, f(x)=(x+1)3(x3)3 is strictly decreasing in the interval (,1) and (1,1).

Question 7: Show that y=log(1+x)2x2+x,x>1 is an increasing function of x throughout its domain.

Answer:

Given function is,
f(x)y=log(1+x)2x2+x
f(x)dydx=11+x2(2+x)(1)(2x)(2+x)2=11+x4+2x2x(2+x)2
=11+x4(2+x)2=(2+x)24(x+1)(x+1)(2+x)2
=4+x2+4x4x4(x+1)(2+x)2=x2(x+1)(2+x)2
f(x)=x2(x+1)(x+2)2
Now, for x>1 , is is clear that f(x)=x2(x+1)(x+2)2>0
Hence, f(x)y=log(1+x)2x2+x strictly increasing when x>1

Question 8: Find the values of x for which y=[x(x2)]2 is an increasing function.

Answer:

Given function is,
f(x)y=[x(x2)]2
f(x)dydx=2[x(x2)][(x2)+x]
=2(x22x)(2x2)
=4x(x2)(x1)
Now,

f(x)=0

4x(x2)(x1)=0

x=0, x=2 and x=1

So, the intervals are (,0), (0,1), (1,2) and (2,)

In interval (0,1) and (2,), f(x)>0
Hence, f(x)y=[x(x2)]2 is an increasing function in the interval (0,1)(2,)

Question 9: Prove that y=4sinθ(2+cosθ)θ is an increasing function of θin[0,π/2]

Answer:

Given function is,
f(x)=y=4sinθ(2+cosθ)θ

f(x)=dydθ=4cosθ(2+cosθ)(sinθ)4sinθ)(2+cosθ)21
=8cosθ+4cos2θ+4sin2θ(2+cosθ)2(2+cosθ)2
=8cosθ+4(cos2θ+sin2θ)4cos2θ4cosθ(2+cosθ)2
=8cosθ+44cos2θ4cosθ(2+cosθ)2
=4cosθcos2θ(2+cosθ)2
Now, for θ ϵ [0,π2]

4cosθcos2θ

4cosθcos2θ0 and (2+cosθ)2>0
So, f(x)>0 for θ in [0,π2]
Hence, f(x)=y=4sinθ(2+cosθ)θ is increasing function in θ ϵ [0,π2]

Question 10: Prove that the logarithmic function is increasing on (0,)

Answer:

Let logarithmic function is log x
f(x)=logx
f(x)=1x
Now, for all values of x in (0,) , f(x)>0
Hence, the logarithmic function f(x)=logx is increasing in the interval (0,)

Question 11: Prove that the function f given by f(x)=x2x+1 is neither strictly increasing nor decreasing on (– 1, 1).

Answer:

Given function is,
f(x)=x2x+1
f(x)=2x1
Now, for interval (1,12) , f(x)<0 and for interval (12,1),f(x)>0
Hence, by this, we can say that f(x)=x2x+1 is neither strictly increasing nor decreasing in the interval (-1,1)

Question 12: Which of the following functions are decreasing on 0,π/2

(A)cosx(B)cos2x(C)cos3x(D)tanx

Answer:

(A)
f(x)=cosxf(x)=sinx
f(x)<0 for x in (0,π2)
Hence, f(x)=cosx is decreasing function in (0,π2)

(B)
f(x)=cos2xf(x)=2sin2x
Now, as
0<x<π20<2x<π
f(x)<0 for 2x in (0,π)
Hence, f(x)=cos2x is decreasing function in (0,π2)

(C)
f(x)=cos3xf(x)=3sin3x
Now, as
0<x<π20<3x<3π2
f(x)<0 for x ϵ (0,π3) and f(x)>0 x ϵ (π3,π2)
Hence, it is clear that f(x)=cos3x is neither increasing nor decreasing in (0,π2)

(D)
f(x)=tanxf(x)=sec2x
f(x)>0 for x in (0,π2)
Hence, f(x)=tanx is strictly increasing function in the interval (0,π2)

So, only (A) and (B) are decreasing functions in (0,π2)

Question 13: On which of the following intervals is the function f given by f(x)=x100+sinx1 decreasing ?
(A) (0,1)

(B) π2,π

(C) 0,π2

(D) None of these

Answer:

(A) Given function is,
f(x)=x100+sinx1
f(x)=100x99+cosx
Now, in interval (0,1)
f(x)>0
Hence, f(x)=x100+sinx1 is increasing function in interval (0,1)

(B) Now, in interval (π2,π)

100x99>0 but cosx<0
100x99>cosx100x99cosx>0 , f(x)>0
Hence, f(x)=x100+sinx1 is increasing function in interval (π2,π)

(C) Now, in interval (0,π2)

100x99>0 and cosx>0
100x99>cosx100x99cosx>0 , f(x)>0
Hence, f(x)=x100+sinx1 is increasing function in interval (0,π2)

So, f(x)=x100+sinx1 is increasing for all cases
Hence, correct answer is None of these

Question 14: For what values of a the function f given by f(x)=x2+ax+1 is increasing on
[1, 2]?

Answer:

Given function is,
f(x)=x2+ax+1
f(x)=2x+a
Now, we can clearly see that for every value of a>2
f(x)=2x+a >0
Hence, f(x)=x2+ax+1 is increasing for every value of a>2 in the interval [1,2]

Question 15: Let I be any interval disjoint from [–1, 1]. Prove that the function f given by f(x)=x+1/x is increasing on I.

Answer:

Given function is,
f(x)=x+1/x
f(x)=11x2
Now,
f(x)=011x2=0x2=1x=±1

So, intervals are from (,1), (1,1) and (1,)
In interval (,1),(1,) , 1x2<111x2>0
f(x)>0
Hence, f(x)=x+1/x is increasing in interval (,1)(1,)
In interval (-1,1) , 1x2>111x2<0
f(x)<0
Hence, f(x)=x+1/x is decreasing in interval (-1,1)
Hence, the function f given by f(x)=x+1/x is increasing on I disjoint from [–1, 1]

Question 16 : Prove that the function f given by f(x)=logsinx is increasing on

(0,π/2)anddecreasingon(π/2,π)


Answer:

Given function is,
f(x)=logsinx
f(x)=1sinxcosx=cotx

Now, we know that cotx is positive in the interval (0,π2) and negative in the interval (π2,π).

f(x)>0 in (0,π2) and f(x)<0 in (π2,π).
Hence, f(x)=logsinx is increasing in the interval (0,π/2) and decreasing in interval (π/2,π)

Question 17: Prove that the function f given by f (x) = log |cos x| is decreasing on (0,π/2)
and increasing on (3π/2,2π)

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,
f(x)=1cosx(sinx)=tanx
We know that in interval (0,π2) , tanx>0tanx<0
f(x)<0
Hence, f(x) = log|cos x| is decreasing in interval (0,π2)

We know that in interval (3π2,2π) , tanx<0tanx>0
f(x)>0
Hence, f(x) = log|cos x| is increasing in interval (3π2,2π)

Question 18: Prove that the function given by f(x)=x33x2+3x100 is increasing in R.

Answer:

Given function is,
f(x)=x33x2+3x100
f(x)=3x26x+3
=3(x22x+1)=3(x1)2
f(x)=3(x1)2
We can clearly see that for any value of x in R f(x)>0
Hence, f(x)=x33x2+3x100 is an increasing function in R

Question 19: The interval in which y=x2ex is increasing is

(A) (,)

(B) (2,0)

(C) (2,)

(D) (0,2)

Answer:

Given function is,
f(x)y=x2ex
f(x)dydx=2xex+ex(x2)
xex(2x)
f(x)=xex(2x)
Now, it is clear that f(x)>0 only in the interval (0,2)
So, f(x)y=x2ex is an increasing function for the interval (0,2)
Hence, (0,2) is the answer


Also, read,

Background wave

Topics covered in Chapter 7 Integrals: Exercise 6.2

Increasing and decreasing function: Let I be an interval contained in the domain of a real-valued function f. Then f is said to be

(i) increasing on I if x1<x2 in If(x1)f(x2) for all x1,x2I.

(ii) decreasing on I, if x1<x2 in If(x1)f(x2) for all x1,x2I.

(iii) constant on I, if f(x)=c for all xI, where c is a constant.

(iv) strictly increasing on I if x1<x2 in If(x1)<f(x2) for all x1,x2I.

(v) strictly decreasing on I if x1<x2 in If(x1)>f(x2) for all x1,x2I.

Also, read,

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JEE Main high scoring chapters and topics

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Unlock chapter-by-chapter guidance of NCERT solutions of other subjects using the links below.

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Subject-Wise NCERT Exemplar Solutions

Tap in these links below for detailed NCERT exemplar solutions for other subjects.

JEE Main Important Mathematics Formulas

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

Frequently Asked Questions (FAQs)

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.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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