Karnataka 2nd PUC Mathematics Application of Derivatives MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: APPLICATIONS OF DERIVATIVES

Rate Measure
dy
can be viewed as the rate of change of y with respect to x.
dx
ds dv d 2 s
(i) velocity v = (ii) acceleration a = 
dt dt dt 2
 If a quantity y varies with another quantity x, satisfying some rule y = f(x),
then dx/dy
 f ′(x) represents the rate of change of y with respect to x and (dy/dx)x=x0
 f′(x0) represents the rate of change of y with respect to x at x = xo.
 If two variables x and y are varying with respect to another variable t, i.e.,
dy
dy dt dx
if x = f(t) and y =g(t) then by Chain Rule :  , if 0
dx dx dt
dt
Increasing & Decreasing
Let f(x) be a real function defined on I = (a, b) or [a, b) or (a, b] or [a, b].
Suppose f is continuous on I and differentiable in (a, b). If
(i) f ′(c) > 0 ∀ c ∈(a, b) then f is strictly increasing on I.
(ii) f ′(c) < 0 ∀ c ∈ (a, b) then f is strictly decreasing on I.
(iii) f ′(c)  0 ∀ c ∈ (a, b) then f is increasing on I.
(iv) f ′(c)  0 ∀ c ∈ (a, b) then f is decreasing on I
 A function f is said to be increasing on an interval (a, b) if x1 < x2 in (a, b)
⇒ f(x1) < f(x2) for all x1, x2 ∈ (a, b).
 A function f is said to be decreasing on an interval (a, b) if x1 < x2 in (a, b)
⇒ f(x1) > f(x2) for all x1, x2 ∈ (a, b).
A function is called monotonic, if it is either increasing or decreasing.
Critical point
A point x = c in the domain of the function f is said to be a critical point of the function f
if either f ′(c) = 0 or f ′(c) does not exist or f is not differentiable
Example: For the function f(x) = x2 - 4x + 6, x = 2 is a critical point
 The critical point x = c of a function f separates the increasing and decreasing parts of
the function if f 11  c   0 .
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Stationary point
A point x = c in the domain of the function is said to be a stationary point of
y =f(x) if f ′(c) = 0.
Example: The stationary points of f(x) = sin x are ± π/2, ±3 π/2, ±5 π/2.....
Maxima and Minima
 The global minimum value is also called absolute minimum or least value
 The global maximum value is also called absolute maximum value or greatest
value
 Relative maximum is also called as local maximum
 Relative minimum is also called as local minimum.
 A turning point of a function f is a point of either relative maximum or relative
minimum.
First derivative test
Given a curve y = f(x),
 For the stationary point at x = a, Second derivative test
dy
 If changes sign from negative to positive as x increases through a, the point S
dx
is a minimum point
dy
 If changes sign from positive to negative as x increases through a, the point S
dx
is a maximum point
dy
 If does not change sign as x increase through a, the point S is a point of
dx
inflexion.
 A stationary point is called a turning point if it is either a maximum point or a
minimum point.
Second Derivative Test
dy dy
(1) Find  0 and solve for x. Suppose one root of  0 is at x=a
dx dx
d2y
(2) If  ve for x=a, then Max. at x=a
dx 2
d2y
If  ve for x=a, then Min. at x=a
dx 2
d2y d3y
(3) If  0 at x=a, then find .
dx 2 dx 3
d3y
If  0 at x=a, then neither maximum nor minimum at x=a
dx 3
d3y d4y
(4) If  0 at x=a, then find .
dx 3 dx 4
d4y
If  ve at x=a (Min Value)
dx 4
d4y
If  ve at x=a (Max Value)
dx 4
d4y d5y
If at x=a,  0 then find and procedure further
dx 4 dx 5

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Note:
 If f 11  a   0 and a is not a point of local maximum or local minimum, then a is a
point of inflection.
 As f is twice differentiable at a, we mean second order derivative of f exists at a.

Note:
 Of all rectangles of a given perimeter p, the square has the largest area, with length of
the side p/4
 All rectangles of a given area A, the squares has the least perimeter, with length of the
side A
 All the rectangles that can be inscribed in a given circle
square has the largest area, whose length of the diagonal is the
 Diagonal 
2

diameter of the circle A = 2×  Radius 
2
=
2

 A cone of maximum volume that can be inscribed in a sphere
4  Radius 
of a given radius r, is of height .
3

 A right circular cylinder of maximum volume that can be
2  Radius 
inscribed in a square of radius r, is of height .
3

 The length and breadth of the greatest rectangle that
x2 y 2
can be inscribed in an ellipse 2 + 2 = 1 are a 2 & b 2
a b

 For any given perimeter, the triangle with the maximum
area is an equilateral triangle

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 The area of an isosceles triangle will be maximum when it is a right angles triangle
1
The maximum area for an isosceles right triangle with equal sides a is A = a 2
2
 The largest isosceles triangle with a given perimeter a is an equilateral triangle, where
a 3
 side 
2
all sides are equal to . The maximum area A is calculated by A =
3 4
2
x + bx + c
 f x = 2  b.c > 0  attains maximum at x = c and minimum at x = - c . The
x - bx + c
   
maximum and minimum values are given by f c & f - c respectively.


Increasing in  c , c 
x 2 - bx + c
 f x =  b.c > 0  attains at minimum x = c and maximum at x = - c . The
x 2 + bx + c
   
minimum and maximum values are given by f c & f - c respectively.


Decreasing in  c , c 
2
ax + bx + c b
 f  x = 2
attains maximum at x = - , if c  c1 . The corresponding maximum
ax + bx + c1 2a
b 2 - 4ac
value is given by 2
b - 4ac1
ax 2 + bx + c b
 f  x = 2 attains minimum at x = - , if c  c1 . The corresponding minimum
ax + bx + c1 2a
b 2 - 4ac
value is given by 2
b - 4ac1
 f  x   ae x  be x  ab  0  has minimum value to 2 ab
 The perimeter of a sector is a constant. If its area is to be maximum, then the sectorial
angle is: 2c
 If sum of two positive numbers is k, then sum of their squares / cubes is minimum
k k
when they are: ,
2 2
nk mk
 If the sum of two numbers is k, then product xn y m maximum will be: ,
n+m n+m
 The product of two positive numbers is k. If the sum of their squares is minimum, then
the numbers are: k , k

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Rate of Change of Quantities:
𝟐𝒕𝟑 𝟓
1) The distance ‘𝒔’ meters travelled by a particle in ‘𝒕’ seconds is given by 𝒔 = 𝟑 − 𝟏𝟖𝒕 + 𝟑.

The acceleration when the particle comes to rest is (Easy) (KCET-2023)
(A) 12 𝑚/𝑠𝑒𝑐 2 (B) 3 𝑚/𝑠𝑒𝑐 2 (C) 18 𝑚/𝑠𝑒𝑐 2 (D) 10 𝑚/𝑠𝑒𝑐 2
𝒙𝟐 𝒚𝟐
2) A Particle moves along the curve 𝟏𝟔 + 𝟒 = 𝟏. When the rate of change of abscissa is 4

times that of its ordinate, then the quadrant in which the particle lies is
(Easy) (KCET-2023)
(A) III or IV (B) I or III (C) II or III (D) II or IV
3) A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the
rate of 0.05 𝒄𝒎/𝒔𝒆𝒄. The rate at which its area is increasing, when the radius is 5.2
cm, is (Easy) (KCET-2023)
(A) 5.05𝜋 𝑐𝑚2 /𝑠𝑒𝑐 (B) 5.2𝜋 𝑐𝑚2 /𝑠𝑒𝑐
(C) 0.52𝜋 𝑐𝑚2 /𝑠𝑒𝑐 (D) 27.4𝜋 𝑐𝑚2 /𝑠𝑒𝑐
4) A drone camera is moving along 𝟑𝒚 = 𝒙𝟑 − 𝟑. When y-coordinate changes 9 times as
fast as x-coordinate, it captures good quality pictures. Then one of the precise
positions of the drone at that instant is (Easy) (KCET-2026)
(A) (-3, 8) (B) (3, -8) (C) (3, 8) (D) (-3, -8)
5) The sides of an equilateral triangle are increasing at the rate of 4 cm/sec. The rate at
which its area is increasing, when the side is 14 cm, is (Easy)
(A) 42 cm2/sec (B) 7√3 cm2/sec (C) 28√3 cm2/sec (D) 14√3 cm2/sec
6) A ladder, 5 meter long, stands on a horizontal floor, leans against a vertical wall. If the
top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the
angle between the floor and the ladder is decreasing when lower end of the ladder is
2 meter from the wall is (Average)
1 1
(A) 10 radians/sec (B) 20 radians/sec

(C) 20 radians/sec (D) 10 radians/sec
7) The rate of change of the surface area of a sphere of radius 𝒓 when the radius is
increasing at the rate of 2 cm/sec is proportional to (Easy)
1 1
(A) 𝑟 2 (B) 𝑟 (C) 𝑟 2 (D) 𝑟

8) If the distance ‘𝒔’ meters traversed by a particle in ‘𝒕’ seconds is given by 𝒔 = 𝒕𝟑 − 𝟑𝒕𝟐 ,
then the velocity of the particle when the acceleration is zero, in m/sec is (Easy)
(A) 3 (B) – 2 (C) – 3 (D) 2

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𝒙𝟐
9) If the rate of decrease of 𝟐 − 𝟐𝒙 + 𝟓 is twice the rate of decrease of x, then x = (Easy)

(A) 1 (B) 2 (C) 3 (D) 4
10) If 𝐬 = 𝟔𝟎𝐭 − 𝟓𝐭 𝟐 denotes the distance covered by a particle in time ' 𝒕 ', then the
distance it covers before coming to rest is ____units. (Easy)
(A) 120 (B) 720 (C) 240 (D) 180
11) If 𝒙 and 𝒚 are the sides of two squares such that 𝒚 = 𝒙 − 𝒙𝟐 , then what is the rate of
change of the area of second square with respect to the area of first square?
(Average)
(A) (1 − 𝑥 2 )𝑥 (B) 2(1 − 𝑥 2 )𝑥 (C) 2𝑥 2 − 3𝑥 + 1 (D) 2(2𝑥 2 − 3𝑥 + 1)
12) A cylindrical tank of radius 2 m is being filled with rice at the rate of 314 cubic 𝐦/𝐡.
The depth of the rice is increasing at the rate of (Average)
3
(A) 25m/h (B) 0.25m/h (C) 1m/h (D) 4 m/h

13) A stone is dropped into a quiet lake and waves move in circles at the speed of 6 cm
per second. At the instant when the radius of the circular wave is 12 cm , the
enclosed area is increasing at the rate of (Easy)
(A) 120𝜋cm2 /s (B) 130𝜋cm2 /s (C) 144𝜋cm2 /s (D) none of these
14) If the distance 𝒔 travelled by a particle in time 𝒕 is given by 𝐬 = 𝐭 𝟐 − 𝟐𝐭 + 𝟓, then its
acceleration is (Easy)
(A) 0 (B) 1 (C) 2 (D) 3
15) The edge of a cube is increasing at the rate of 5 cm/sec How fast is the volume of the
cube increasing when the edge is 12 cm long (Easy)
(A) 432 cm3 / sec (B) 2160 cm3 / sec (C) 180 cm3 / sec (D) 53  12 cm3 / sec
16) Assertion (A): A particle moving in a straight line covers a distance of x cm in t
seconds, where x  t 3  3t 2  6t  18 . The velocity of the particle at the end of 3 seconds is
9cm/s
𝑑𝑥
Reason (R): Velocity of the particle at the end of 3 seconds is 𝑑𝑡 𝑎𝑡 𝑡 = 3 (Difficult)

(A) A is false and R is true (B) A is true and R is false
(C) A is true and R is true (D) A is false and R is false
17) A point on the curve 𝟑𝒚 = 𝟔𝒙 − 𝒙𝟑 at which the rate of change of ordinate is equal to
the rate of change of abscissa, is (Average)
5 5 4
(A) (1, 3) (B) (2, 3) (C) (1, 2) (D) (2, 3)

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18) On the curve 𝒙𝟑 = 𝟏𝟐𝒚 the abscissa changes at a faster rate than the ordinate. Then x
belongs to the interval (Average)
(A) (−2, 2) (B) (−3, 0) (C) (0, 3) (D) (2, 3)
19) A water tank has the shape of an inverted right circular cone, whose semi-vertical
𝟏
angle is 𝐭𝐚𝐧−𝟏 (𝟑). Water is poured into it at a constant rate of 10 cubic cm/sec. Then

the rate (in cm/sec), at which the level of water is rising at the instant when the
depth of water in the tank is 5 cm, is (Difficult)
9 18 3 1
(A) 5𝜋 (B) 5𝜋 (C) 5𝜋 (D) 5𝜋

Rate of Change of Quantit
1 2 3 4 5 6 7 8 9 10
A D C C C B D C D D
11 12 13 14 15 16 17 18 19 20
C A C C B A A A B

Increasing and decreasing Functions:
1) The function 𝒙𝒙 , 𝒙 > 𝟎 is increasing at (Average) (KCET-2024)
1 1
(A) ∀ 𝑥 ∈ 𝑅 (B) 𝑥 < 𝑒 (C) 𝑥 > 𝑒 (D) 𝑥 < 0

2) If 𝒇(𝒙) = 𝒙𝒆𝒙(𝟏−𝒙) then 𝒇(𝒙) is (Difficult) (KCET-2024)
(A) increasing in R (B) decreasing in R
1 1
(C) decreasing in (− 2 , 1) (D) increasing in (− 2 , 1)

3) The function 𝒇(𝒙) = 𝐭𝐚𝐧 𝒙 − 𝒙 is (Average) (KCET-2025)
(A) always increasing (B) always decreasing
(C) never increasing (D) neither increasing nor decreasing
4) The interval in which the function 𝒇(𝒙) = 𝒙𝟑 − 𝟔𝒙𝟐 + 𝟗𝒙 + 𝟏𝟎 is increasing is (Average)
(A) [1, 3] (B) (−∞, 1) ∪ (3, ∞) (C) (−∞, 1) ∪ [3, ∞) (D) (−∞, 1] ∪ [3, ∞)
5) The function 𝒇(𝒙) = 𝒙 + 𝒄𝒐𝒔𝒙, 𝒙 ∈ [𝟎, 𝝅] is (Average)
(A) increasing (B) decreasing
(C) constant (D) 𝑓 ′ (𝑥) > 0, ∀ 𝑥 ∈ [0, 𝜋]
6) The value of 𝒙 for which 𝟐𝒙𝟑 − 𝟗𝒙𝟐 + 𝟏𝟐𝐱 + 𝟐 is decreasing (Easy)
(A) 1 < 𝑥 < 2 (B) 𝑥 < 1 𝑜𝑟 𝑥 > 2
(C) −2 < 𝑥 < −1 (D) 𝑥 < −2 𝑜𝑟 𝑥 > −1
7) The function 𝐟(𝐱) = 𝐜𝐨𝐭 −𝟏 𝐱 + 𝐱 is (Average)
A) neither increasing nor decreasing on R (B) constant on R
C) increasing on R (D) decreasing on R
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𝟏
8) The function 𝒇(𝒙) = 𝟏+𝒙𝟐 (Average)

(A) increases for all real 𝑥 (B) decreases for all real 𝑥
(C) increases for all positive real 𝑥. (D) decreases for all positive real 𝑥.
2 x
9) The function y  x e is decreasing on (Easy)
(A)(−∞, 2) (B)(−∞, 0) ∪ (2, ∞) (C)(0, ∞) (D)(0, 2)
10) The function f given by f (x) = 𝒙𝟐 – x + 1 is. (Easy)
1
A) neither increasing nor decreasing on (– 1, 1) B) decreasing on (2 , ∞)
1
C) increasing on (–∞,∞) D) increasing on (−∞, 2)

11) The function f given by 𝒇(𝒙) = 𝒍𝒐𝒈|𝒔𝒊𝒏𝒙|, 𝒙 ≠ 𝒏𝝅, 𝒏 ∈ 𝒁, is increasing on (Average)
π π 3π
A) (0, 2 ) B)(0, π) C)( 2 , π) D)( 2 , 2π)

12) The function 𝒇(𝒙) = 𝟑𝒙𝟑 − 𝟑𝟔𝒙 + 𝟗𝟗 is increasing for (Average)
(A) −∞ < 𝑥 < 2 (B) −2 < 𝑥 < ∞ (C) −2 < 𝑥 < 2 (D) (−∞, −2) ∪ (2, ∞)
13) Let 𝒇(𝒙) = 𝒔𝒊𝒏𝟐 𝒙 + 𝟑𝒄𝒐𝒔𝒙. Then 𝒇 is decreasing in (Average)
𝜋 3𝜋 3𝜋
(A) (0, 𝜋) (B) (2 , ) (C) (𝜋, 2𝜋) (D) ( 2 , 2𝜋)
2

14) The function 𝒇(𝒙) = 𝐭𝐚𝐧−𝟏 (𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙) is an increasing function in (Difficult)
𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋
(A) ( 4 , 2 ) (B) (− 2 , 2 ) (C) (0, 2 ) (D) (− 2 , 4 )

15) 𝒚 = 𝒙(𝒙 − 𝟑)𝟐 decreases for the values of 𝒙 given by (Average)
3
(A) 1 < 𝑥 < 3 (B) 𝑥 < 0 (C) 𝑥 > 0 (D) 0 < 𝑥 < 2

16) The function 𝒚 = 𝟑𝒙𝟓 − 𝒙𝟔 is increasing on (Average)
5 5
(A) (0, 2) (B) (2 , 3) (C) (−3, 3) (D) (0, 3)

17) The function𝒇(𝒙) = 𝒙𝟐 − 𝟐𝒂𝒙 is increasing in (𝟒, ∞), then maximum value of ‘𝒂’ is
(Difficult)
(A) 3 (B) 4 (C) 5 (D) 0
18) The function 𝒇(𝒙) = 𝒄𝒐𝒔𝟑 𝒙 − 𝟑𝒄𝒐𝒔𝟐 𝒙 + 𝟏𝟖𝒄𝒐𝒔𝒙 + 𝟐𝟎 is (Difficult)
𝜋 3𝜋
(A) increasing on (0, 2 ) (B) decreasing on (𝜋, 2 )
3𝜋 𝜋
(C) decreasing on ( 2 , 2𝜋) (D) decreasing on ( 2 , 𝜋)

19) Statement 1: The function 𝑓(𝑥) = tan−1 𝑥 is an increasing function.
Statement 2: A function 𝑓(𝑥) is increasing if 𝑓 ′ (𝑥) > 0, ∀ 𝑥 in its domain. (Average)
(A) Statement 1 is true and statement 2 is false
(B) Statement 1 is false and statement 2 is true
(C) Both the statements are false (D) Both the statements are true

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𝟏
20) If 𝟑𝒇(𝒙) + 𝟐𝒇 (𝒙) = 𝒙𝟐 − 𝟑, ∀ 𝒙 ≠ 𝟎 and 𝒚 = 𝟓𝒙𝟐 𝒇(𝒙), then 𝒚 is increasing in (Difficult)
1 1 1 1
(A) (− , 0) ∪ ( , ∞) (B) (−∞, − ) ∪ (0, )
√2 √2 √2 √2
1 1 1 1
(C) (−∞, − )∪( , ∞) (D) (− , 0) ∪ (0, )
√2 √2 √2 √2

Increasing and decreasing Functions
1 2 3 4 5 6 7 8 9 10
C D A B A A C D B A
11 12 13 14 15 16 17 18 19 20
A D A D A A B D D A

Maxima and Minima:
Local Maxima and Local Minima:
1) An enemy fighter jet is flying along the curve given by 𝒚 = 𝒙𝟐 + 𝟐. A solider is placed
at (3, 2) wants to shoot down the jet when it is nearest to him. Then the nearest
distance is (Difficult) (CET-2023)
(A) 2 units (B) √3 units (C) √5 units (D) √6 units
2) The maximum volume of a right circular cone with slant height 6 units is
(Difficult) (CET-2024)
(A) 4√3𝜋 cubic units (B) 16√3𝜋 cubic units
(C) 3√3𝜋 cubic units (D) 6√3𝜋 cubic units
3) For the function 𝒇(𝒙) = 𝒙𝟑 − 𝟔𝒙𝟐 + 𝟏𝟐𝒙 − 𝟑, 𝒙 = 𝟐 is (Average) (CET-2024)
(A) a point of minimum (B) a point of inflexion
(C) not a critical point (D) a point of maximum
4) The minimum value of 𝟏 − 𝐬𝐢𝐧 𝒙 is (Easy) (CET-2025)
(A) 0 (B) – 1 (C) 1 (D) 2
𝒙 𝝅
5) The function 𝒇(𝒙) = 𝟏+𝒙 𝒕𝒂𝒏𝒙 in (𝟎, 𝟐 ), has (Difficult)

(A) a minimum value (B) a maximum value
(C) no extreme value (D) both maximum value and minimum value
6) If 𝒚 = 𝒂 𝒍𝒐𝒈𝒙 + 𝒃𝒙𝟐 + 𝐱 has its extreme values at 𝒙 = −𝟏 and 𝒙 = 𝟐 then (Average)
1
A) 𝑎 = 2, 𝑏 = −1 B) 𝑎 = −2, 𝑏 = 2
1 1
C) 𝑎 = 2, 𝑏 = − 2 D) 𝑎 = −2, 𝑏 = − 2

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𝝅 𝝅
7) The maximum value of the function 𝒇(𝒙) = 𝐬𝐢𝐧 (𝒙 + 𝟔 ) + 𝐜𝐨𝐬 (𝒙 + 𝟔 ) in the interval
𝝅
(𝟎, 𝟐 ) occurs at (Average)
𝜋 𝜋 𝜋 𝜋
A) 12 B) 6 C) 4 D) 3

8) The minimum distance of the point (𝟎, 𝟖) from the curve 𝒚 = 𝒙𝟐 is (Difficult)
√29 √33 √31 √27
(A) (B) (C) (D)
2 2 2 2

9) If 𝒚 = (𝒙 − 𝒂)𝟐+(𝒙 − 𝒃)𝟐 + (𝒙 − 𝒄)𝟐 , then y is minimum when x is equal to (Average)
𝑎+𝑏+𝑐 𝑎+𝑏+𝑐
(A) (B) (C) 𝑎 + 𝑏 + 𝑐 (D) 2(𝑎 + 𝑏 + 𝑐)
2 3

10) A population of p(t) of 1000 bacteria introduced into nutrient medium grows
𝟏𝟎𝟎𝟎𝒕
according to the relation 𝒑(𝒕) = 𝟏𝟎𝟎𝟎 + 𝟏𝟎𝟎+𝒕𝟐. The maximum size of this bacterial

population is (Average)
(A) 1100 (B) 1250 (C) 1050 (D) 5250
11) The perimeter of a sector of a circle is 𝒑. The area of the sector is maximum when its
radius is (Average)
𝑝 1 𝑝
(A) 4 (B) √𝑝 (C) (D) 2
√𝑝
𝒍𝒐𝒈𝒙
12) The maximum value of , 𝒙 ∈ (𝟎, ∞)is (Easy)
𝒙
1 2
(A) 𝑒 (B) 𝑒 (C) e (D) 1

13) If 𝒙 + 𝒚 = 𝟏𝟐 then the minimum value of 𝒙𝟐 + 𝒚𝟐 is (Easy)
(A) 72 (B) 144 (C) 48 (D) 36
14) Find the volume of the largest cylinder that can be inscribed in a sphere of radius 𝒓 cm.
(Difficult)
𝜋𝑟 3 4𝜋𝑟 2 ℎ 4𝜋𝑟 3
(A) 3√3 (B) (C) 4𝜋𝑟 3 (D) 3√3
3√3

15) Twenty meters of wire is available for fencing off a flower bed in the form of a circular
sector. Then the maximum area (in sq. m) of the flower bed is (Average)
(A) 10 (B) 25 (C) 30 (D) 12.5
16) The number of points of local maxima and local minima of the function
𝒇(𝒙) = 𝟑𝒙𝟑 + 𝟔𝒙𝟐 + 𝟒𝒙 + 𝟕 is (Average)
(A) 1 (B) 2 (C) 3 (D) 0
𝟏
17) Consider the function 𝒇(𝒙) = 𝒙 + 𝒙, 𝒙 ≠ 𝟎, then (Average)

(A) the local maximum value of 𝑓(𝑥) < the local minimum value of 𝑓(𝑥)
(B) the local maximum value of 𝑓(𝑥) > the local minimum value of 𝑓(𝑥)
(C) 𝑓(𝑥) has no local minimum value (D) 𝑓(𝑥) has no local maximum value
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18) Let 𝒇: 𝑹 → 𝑹 be defined by 𝒇(𝒙) = 𝟐𝒙 − 𝒔𝒊𝒏𝒙, then 𝒇 (Easy)
𝜋
(A) has a minimum at 𝑥 = 2 (B) has a maximum at 𝑥 = 0

(C) is a decreasing function (D) is an increasing function
19) Assertion (A): The function 𝑓(𝑥) = 𝑥 3 has no local maxima at 𝑥 = 0.
Reason (R): A differentiable function 𝑓(𝑥) does not have a local maxima at a point 𝑥 = 𝑎, if
𝑓 ′ (𝑥) does not change sign as 𝑥 increases through ‘𝑎’. (Easy)
(A) A is false and R is true (B) A is true and R is false
(C) A is true and R is true (D) A is false and R is false
20) 𝒇(𝒙) = (𝟐𝒙)𝒙 , has a stationary point at (Average)
1 1
(A) 𝑥 = 𝑒 2 (B) 𝑥 = 𝑒 (C) 𝑥 = 𝑒 (D) 𝑥 = 2𝑒

21) The maximum value of 𝒇(𝒙) = (𝟑𝒙)−𝒙 , is (Average)
1 1
(A) 𝑒 −3𝑒 (B) 𝑒 3𝑒 (C) 𝑒 3𝑒 (D) (3𝑒)𝑒
22) For the function 𝒚 = 𝒇(𝒙) given in the figure, consider the following statements 1 and 2

Statement 1: At 𝑥 = 0, the function 𝑦 = 𝑓(𝑥) has local minima.
Statement 2: The function 𝑦 = 𝑓(𝑥) has no local maxima. (Easy)
(A) Statement 1 is false and statement 2 is true
(B) Statement 1 is true and statement 2 is false
(C) Both the statements are false
(D) Both the statements are true
𝒏
23) The largest term in the sequence 𝒂𝒏 = 𝒏𝟐 +𝟏𝟎𝟎, (𝒏 ∈ 𝑵), is (Average)

(A) 𝑎5 (B) 𝑎7 or 𝑎8 (C) 𝑎4 (D) 𝑎10

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Absolute Maximum values and Absolute Minimum values:
24) Assertion (A): The maximum value of the function f(x) = 𝑥 3 , x ∈[-1, 1], is attained at its end
point, x = 1.
Reason (R): The function f(x) = 𝑥 3 is an increasing function in [-1, 1]. (Easy)
A) Both A and R are true and R is the correct explanation for A
B) Both A and R are true but R is not the correct explanation for A.
C) A is false and R is true.
D) Both A and R are false.
25) The absolute maximum value of y = 𝒙𝟑 – 3x + 2 in 0 ≤ x ≤ 2 is (Average)
A) 0 B) 2 C) 4 D) 6
26) The smallest value of the polynomial 𝒙𝟑 − 𝟏𝟖𝒙𝟐 + 𝟗𝟔𝒙 in [𝟎, 𝟗] is (Average)
(A) 128 (B) 0 (C) 135 (D) 160
27) The absolute maximum value of the function 𝒇(𝒙) = 𝟐𝒙𝟑 − 𝟗𝒙𝟐 + 𝟏𝟐𝒙 + 𝟏 on [𝟎, 𝟐] is
(Average)
(A) 8 (B) 1 (C) 6 (D) 5
28) For the function 𝒚 = 𝒇(𝒙), 𝒙 ∈ [−𝟐, 𝟐] given in the figure, consider the following
statements 1 and 2

Statement 1: The absolute maximum value of 𝑦 = 𝑓(𝑥), 𝑥 ∈ [−2, 2] is 𝑓(2).
Statement 2: The function 𝑦 = 𝑓(𝑥), 𝑥 ∈ [−2, 2] is increasing in [−2, 0]. (Average)
(A) Statement 1 is true and statement 2 is false
(B) Statement 1 is false and statement 2 is true
(C) Both the statements are false
(D) Both the statements are true

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29) For the function 𝒚 = 𝒇(𝒙) given in the figure,
consider the following statements 1 and 2.

𝜋 𝜋 𝑑𝑦
Statement 1: In the interval (− 2 , − 6 ) for the above function 𝑑𝑥 < 0.
𝜋
Statement 2: The function 𝑦 = 𝑓(𝑥) has a local maxima at 𝑥 = 2 (Average)

(A) Statement 1 is true and statement 2 is false
(B) Statement 1 is false and statement 2 is true
(C) Both the statements are false
(D) Both the statements are true
30) Let 𝒇(𝒙) = √𝒙 − 𝟐 + √𝟑 − 𝒙 be a real valued function. If 𝜶 and 𝜷 are respectively the
minimum and maximum values of 𝒇, then 𝜶𝟐 + 𝜷𝟐 is equal to (Difficult)
(A) 1 (B) 2 (C) 0 (D) 3
𝝅 𝝅
31) Find the maximum value of 𝒇(𝒙) = 𝒔𝒊𝒏𝟑𝒙 + 𝟒, 𝒙 ∈ [− 𝟗 , 𝟗 ]. (Average)
√3+8 √3−8 8−√3
(A) (B) (C) (D) √3 + 8
2 2 2

Maxima and Minima
1 2 3 4 5 6 7 8 9 10
C B B A B C A C B C
11 12 13 14 15 16 17 18 19 20
A A A D B D A D C D
21 22 23 24 25 26 27 28 29 30
C B D A C B C A A D
31
A

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