Applications of Derivatives: JEE Mains Previous Year Questions (2021-2024)

# Applications of Derivatives: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Download, print and study this document offline
``` Page 1

JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
Page 2

JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Q1 (2)
Q2 (3)
Page 3

JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)

Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2

?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
Page 4

JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)

Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2

?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
?? '
( ?? )= 3 ×
1
3
· ?? '
(
?? 3
)- ?? '
( 3 - ?? )
= ?? '
(
?? 3
)- ?? '
( 3 - ?? )
If g is decreasing in ( 0, ?? )
g
'
( x)< 0
f
'
(
x
3
)+ f
'
( 3 - x)< 0
f
'
(
x
3
)< f
'
( 3 - x)
?
x
3
< 3 - x
? x <
9
4

Therefore ?? =
9
4

Then 8?? = 8 ×
9
4
= 18
Q3
?? ( ?? )= 2?? + 3( ?? )
2
3
?? '
( ?? )= 2 + 2?? -1
3
= 2 (1 +
1
?? 1
3
)
= 2 (
?? 1
3
+ 1
?? 1
3
)

So, maxima ( M) at x = -1&minima ( m) at x = 0
Q4
?? ( ?? )=
?? ?? 2
- 6?? - 16

Now,
Page 5

JEE Mains Previous Year Questions
(2021-2024): Application of
Derivatives
2024
Q1 - 2024 (01 Feb Shift 1)
If 5f( x)+ 4f (
1
x
)= x
2
- 2, ?x ? 0 and y = 9x
2
f( x) , then y is strictly increasing in :
(1) ( 0,
1
v5
)? (
1
v5
, 8)
(2) ( -
1
v5
, 0)? (
1
v5
, 8)
(3) ( -
1
v5
, 0)? ( 0,
1
v5
)
(4) ( -8,
1
v5
)? ( 0,
1
v5
)
Q2 - 2024 (27 Jan Shift 2)
Let ?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0 for all x ? ( 0,3) . If g is decreasing in ( 0, ?? )
and increasing in ( ?? , 3) , then 8?? is
(1) 24
(2) 0
(3) 18
(4) 20
Q3 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )= 2?? + 3( ?? )
2
3
, ?? ? R, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima
Q4 - 2024 (29 Jan Shift 2)
The function ?? ( ?? )=
?? ?? 2
-6?? -16
, ?? ? R - {-2,8}
(1) decreases in ( -2,8) and increases in
( -8, -2)? ( 8, 8)
(2) decreases in ( -8, -2)? ( -2,8)? ( 8, 8)
(3) decreases in ( -8, -2) and increases in ( 8, 8)
(4) increases in ( -8, -2)? ( -2,8)? ( 8, 8)
Q5 - 2024 (30 Jan Shift 1)
Let g: R ? R be a non constant twice differentiable such that g
'
(
1
2
)= g
'
(
3
2
) . If a real
valued function f is defined as ?? ( ?? )=
1
2
[?? ( ?? )+ ?? ( 2 - ?? ) ], then
(1) ?? ''
( ?? )= 0 for atleast two ?? in ( 0,2)
(2) ?? ''
( ?? )= 0 for exactly one ?? in ( 0,1)
(3) ?? ''
( ?? )= 0 for no ?? in ( 0,1)
(4) f
'
(
3
2
)+ f
'
(
1
2
)= 1
Q6 - 2024 (30 Jan Shift 2)
Let ?? ( ?? )= ( ?? + 3)
2
( ?? - 2)
3
, ?? ? [-4,4]. If ?? and ?? are the maximum and minimum
values of ?? , respectively in [-4,4], then the value of ?? - ?? is :
(1) 600
(2) 392
(3) 608
(4) 108
Q1 (2)
Q2 (3)
Q3 (3)
Q4 (2)
Q5 (1)
Q6 (3)

Solutions
Q1
5f( x)+ 4f (
1
?? )= x
2
- 2, ??? ? 0
Substitute ?? ?
1
??
5?? (
1
?? )+ 4?? ( ?? )=
1
?? 2
- 2
On solving (1) and (2)
?? ( ?? )=
5?? 4
- 2?? 2
- 4
9?? 2

?? = 9?? 2
?? ( ?? )
?? = 5?? 4
- 2?? 2
- 4.
????
????
= 20?? 3
- 4??
for strictly increasing
dy
dx
> 0
4x( 5x
2
- 1)> 0
x ? ( -
1
v5
, 0)? (
1
v5
, 8)
Q2
?? ( ?? )= 3?? (
?? 3
)+ ?? ( 3 - ?? ) and ?? ''
( ?? )> 0??? ? ( 0,3)? f
'
( x) is increasing function
?? '
( ?? )= 3 ×
1
3
· ?? '
(
?? 3
)- ?? '
( 3 - ?? )
= ?? '
(
?? 3
)- ?? '
( 3 - ?? )
If g is decreasing in ( 0, ?? )
g
'
( x)< 0
f
'
(
x
3
)+ f
'
( 3 - x)< 0
f
'
(
x
3
)< f
'
( 3 - x)
?
x
3
< 3 - x
? x <
9
4

Therefore ?? =
9
4

Then 8?? = 8 ×
9
4
= 18
Q3
?? ( ?? )= 2?? + 3( ?? )
2
3
?? '
( ?? )= 2 + 2?? -1
3
= 2 (1 +
1
?? 1
3
)
= 2 (
?? 1
3
+ 1
?? 1
3
)

So, maxima ( M) at x = -1&minima ( m) at x = 0
Q4
?? ( ?? )=
?? ?? 2
- 6?? - 16

Now,
f
'
( x)=
-( x
2
+ 16)
( x
2
- 6x - 16)
2

f
'
( x)< 0
Thus ?? ( ?? ) is decreasing in
( -8, -2)? ( -2,8)? ( 8, 8)

Q5
?? '
( ?? )=
?? '
( ?? )- ?? '
( 2 - ?? )
2
, ?? '
(
3
2
)=
?? '
(
3
2
)- ?? '
(
1
2
)
2
= 0
Also f
'
(
1
2
)=
g
'
(
1
2
) -g
'
(
3
2
)
2
= 0, f
'
(
1
2
)= 0
? f
'
(
3
2
)= f
'
(
1
2
)= 0
? roots in (
1
2
, 1) and ( 1,
3
2
)
? f
''
( x) is zero at least twice in (
1
2
,
3
2
)
Q6
f
'
( x)= ( x + 3)
2
· 3( x - 2)
2
+ ( x - 2)
3
2( x + 3)
= 5( x + 3) ( x - 2)
2
( x + 1)
f
'
( x)= 0, x = -3, -1,2

?? ( -4)= -216
?? ( -3)= 0, ?? ( 4)= 49 × 8 = 392
?? = 392, ?? = -216
?? - ?? = 392 + 216 = 608
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Applications of Derivatives: JEE Mains Previous Year Questions (2021-2024) - Mathematics (Maths) for JEE Main & Advanced

 1. What are some common applications of derivatives in real life?
Ans. Some common applications of derivatives in real life include determining maximum and minimum values, analyzing rates of change, optimizing functions, and modeling physical systems such as motion and growth.
 2. How are derivatives used in economics and finance?
Ans. In economics and finance, derivatives are used to analyze and predict changes in variables like stock prices, interest rates, and currency exchange rates. They help in risk management, portfolio optimization, and pricing financial instruments.
 3. Can derivatives be used to solve optimization problems?
Ans. Yes, derivatives can be used to solve optimization problems by finding the critical points of a function and determining whether they correspond to a maximum, minimum, or saddle point. This is useful in maximizing profits, minimizing costs, and optimizing processes.
 4. What is the relationship between derivatives and tangents in calculus?
Ans. Derivatives represent the slope of a function at a given point, which is equivalent to the slope of the tangent line to the function at that point. This relationship allows us to approximate the behavior of a function near a specific point.
 5. How do derivatives help in analyzing motion and velocity?
Ans. Derivatives help in analyzing motion and velocity by providing information about the rate at which an object's position changes with respect to time. The derivative of the position function gives the velocity function, which describes the object's speed and direction at any given moment.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

### Up next

 Explore Courses for JEE exam
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;