Applications of Derivatives

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1844
The least value of $f(x) = x^3 - 12x$, $x \in [0, 3]$ is
(A) $ -16$
(B) $ -9$
(C) $ 0$
(D) $ 16$
MEDIUM
2 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1702
For $f(x)=x+\frac{1}{x} (x \ne 0)$
(A) local maximum value is 2
(B) local minimum value is -2
(C) local maximum value is -2
(D) local minimum value < local maximum value
MEDIUM
3 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1701
The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is:
(A) $400\pi~cm^{3}/cm$
(B) $100\pi~cm^{3}/cm$
(C) $50\pi~cm^{3}/cm$
(D) $25\pi~cm^{3}/cm$
MEDIUM
4 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1700
The surface area of a sphere when its volume changes at the same rate as its radius is :
(A) $4\pi$ sq. units
(B) 1 sq. unit
(C) 4 sq. units
(D) $\pi$ sq. units
MEDIUM
5 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1699
Absolute minimum value of $f(x)=(x-2)^{2}+5$ in the interval [-3, 2] is :
(A) -3
(B) 2
(C) 5
(D) 30
MEDIUM
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#620
The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in [0, 2] is:
(A) 0
(B) 2
(C) 4
(D) 5
MEDIUM
7 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#619
The function \(f(x)=x^{2}-4x+6\) is increasing in the interval
(A) \((0, 2)\)
(B) \((-\infty, 2]\)
(C) \([1, 2]\)
(D) \([2, \infty)\)
MEDIUM
8 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#618
A cylindrical tank of radius \(10\) cm is being filled with sugar at the rate of \(100~\pi~cm^{3}/s\). The rate, at which the height of the sugar inside the tank is increasing, is:
(A) \(0.1~cm/s\)
(B) \(0.5~cm/s\)
(C) \(1~cm/s\)
(D) \(1.1~cm/s\)
MEDIUM
9 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#617
The values of \(\lambda\) so that \(f(x)=\sin x-\cos x-\lambda x+C\) decreases for all real values of x are:
(A) \(1\lt\lambda\lt\sqrt{2}\)
(B) \(\lambda\ge1\)
(C) \(\lambda\ge\sqrt{2}\)
(D) \(\lambda\lt1\)
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#616
If \(f(x)=2x+\cos x\), then f(x):
(A) has a maxima at \(x=\pi\)
(B) has a minima at \(x=\pi\)
(C) is an increasing function
(D) is a decreasing function
MEDIUM
11 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#615
The slope of the curve \(y=-x^{3}+3x^{2}+8x-20\) is maximum at:
(A) (1,-10)
(B) (1,10)
(C) (10, 1)
(D) (-10, 1)
MEDIUM
12 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#614
Let \(f(x)=|x|\), \(x\in R\). Then, which of the following statements is **incorrect**?
(A) f has a minimum value at \(x=0\).
(B) f has no maximum value in R.
(C) f is continuous at \(x=0\).
(D) f is differentiable at \(x=0\).
MEDIUM
13 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#613
A spherical ball has a variable diameter \(\frac{5}{2}(3x+1).\) The rate of change of its volume w.r.t. x, when \(x=1\), is :
(A) \(225\pi\)
(B) \(300\pi\)
(C) \(375\pi\)
(D) \(125\pi\)
MEDIUM
14 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#612
Let \(f(x)\) be a continuous function on [a, b] and differentiable on (a, b). Then, this function \(f(x)\) is strictly increasing in (a, b) if
(A) \(f^{\prime}(x)\lt;0\), \(\forall x\in(a,b)\)
(B) \(f^{\prime}(x)\gt;0\), \(\forall x\in(a,b)\)
(C) \(f^{\prime}(x)=0\), \(\forall x\in(a,b)\)
(D) \(f(x)\gt;0\), \(\forall x\in(a,b)\)
EASY
15 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#611
The function \(f(x)=x^{3}-3x^{2}+12x-18\) is:
(A) strictly decreasing on R
(B) strictly increasing on R
(C) neither strictly increasing nor strictly decreasing on R
(D) strictly decreasing on \((-\infty, 0)\)
MEDIUM
16 Competency APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#610
If the sides of a square are decreasing at the rate of \(1.5~cm/s\) the rate of decrease of its perimeter is:
(A) \(1.5~cm/s\)
(B) \(6~cm/s\)
(C) \(3~cm/s\)
(D) \(2.25~cm/s\)
MEDIUM
17 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#609
The function \(f(x)=kx-\sin~x\) is strictly increasing for
(A) \(k\gt1\)
(B) \(k\lt1\)
(C) \(k\gt-1\)
(D) \(k\lt-1\)
MEDIUM
18 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#608
The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minima at x equal to:
(A) 2
(B) 1
(C) 0
(D) -2
MEDIUM
19 Competency APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#607
Given a curve \(y=7x-x^{3}\) and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when \(x=5\) is:
(A) \(-60~units/sec\)
(B) \(60~units/sec\)
(C) \(-70~units/sec\)
(D) \(-140~units/sec\)
HARD
20 Competency APPLY MCQ_SINGLE 1 Mark(s) 2023
#800
If f(x)=a(x-cos\~x) is strictly decreasing in R, then 'a' belongs to
(A) {0}
(B) (0,∞)
(C) (-∞,0)
(D) (-∞,∞)
MEDIUM
21 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#799
The interval in which the function f(x)=2x³+9x²+12x-1 is decreasing, is
(A) (-1, ∞)
(B) (-2,-1)
(C) (-∞, -2)
(D) [-1, 1]
MEDIUM

Very Short Answer (2 Mark)

1 Competency APPLY VSA 2 Mark(s) 2026 AISSCE(Board Exam)
#1758
Determine the values of x for which $f(x)=\frac{x-3}{x+1}$, $x\ne -1$ is an increasing function.
MEDIUM
2 Competency APPLY VSA 2 Mark(s) 2026 AISSCE(Board Exam)
#1757
Find the values of x for which $f(x)=x^{x}$, $x>0$ is increasing.
MEDIUM
3 Competency APPLY VSA 2 Mark(s) 2026 AISSCE(Board Exam)
#1756
Find the sub-interval(s) of $(0,\frac{\pi}{2})$ in which $f(x)=\tan x-4x$ is increasing.
MEDIUM
4 Competency APPLY VSA 2 Mark(s) 2026 AISSCE(Board Exam)
#1755
If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.
MEDIUM
5 Competency APPLY VSA 2 Mark(s) 2026 AISSCE(Board Exam)
#1754
Find the absolute maximum value of $f(x)=\cos x+\sin^{2}x$, $x \in [0,\pi]$.
MEDIUM
6 Competency APPLY VSA 1 Mark(s) 2026 AISSCE(Board Exam)
#1496
A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm3/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is $\frac{\pi}{6}$
MEDIUM
7 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1446
Determine the values of x for which $f(x)=\frac{x-4}{x+1}$, $x\ne-1$ is an increasing or a decreasing function.
MEDIUM
8 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1422
Surface area of a balloon (spherical), when air is blown into it, increases at a rate of $5\text{ mm}^{2}/\text{s}$. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
MEDIUM
9 Competency UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1405
For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?
MEDIUM
10 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1401
If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasing function.
EASY
11 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1400
Find the least value of 'a' so that $f(x)=2x^{2}-ax+3$ is an increasing function on $[2, 4]$.
MEDIUM
12 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1380
Find the values of 'a' for which $f(x)=\sin x-ax+b$ is increasing on R.
MEDIUM
13 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1359
Find the intervals in which function $f(x)=5x^{\frac{3}{2}}-3x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.
MEDIUM
14 Competency UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1336
The area of the circle is increasing at a uniform rate of $2~cm^{2}/sec$. How fast is the circumference of the circle increasing when the radius $r=5$ cm?
MEDIUM
15 Competency UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1315
The volume of a cube is increasing at the rate of $6~cm^{3}/s.$ How fast is the surface area of cube increasing, when the length of an edge is 8 cm?
MEDIUM
16 Knowledge APPLY VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1314
Find the interval in which the function $f(x)=x^{4}-4x^{3}+10$ is strictly decreasing.
MEDIUM
17 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1290
Show that the function $f(x)=4x^{3}-18x^{2}+27x-7$ has neither maxima nor minima.
MEDIUM
18 Knowledge ANALYZE VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1272
Show that $f(x)=e^{x}-e^{-x}+x-tan^{-1}x$ is strictly increasing in its domain.
MEDIUM
19 Knowledge REMEMBER VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1270
If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$
MEDIUM
20 Knowledge UNDERSTAND VSA 2 Mark(s) 2023
#932
If \(f(x)=a(\tan x-\cot x)\), where \(a>0\), then find whether \(f(x)\) is increasing or decreasing function in its domain.
MEDIUM
21 Knowledge UNDERSTAND VSA 2 Mark(s) 2023
#931
Find the maximum and minimum values of the function given by \(f(x)=5+\sin 2x\).
EASY
22 Knowledge APPLY VSA 2 Mark(s) 2023
#907
Show that the function $f(x)=\frac{16\sin x}{4+\cos x}-x$, is strictly decreasing in $(\frac{\pi}{2},\pi)$
MEDIUM

Short Answer (3 Marks)

1 Competency APPLY SA 3 Mark(s) 2026 CBSE Sample QP
#2089
The circular area of vision on the ground by a surveillance drone fitted with camera increases at the rate of $44 𝑚^2$ per metre as it gains vertical height. The radius of the circle on the ground increases at the rate of 6 m/s. Find the rate at which the height of the drone increases with time, at the instant when the radius of the circular area on the ground is 14 m. (𝑈𝑠𝑒 𝜋 =22/7)
HARD
2 Competency APPLY SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1776
A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.
MEDIUM
3 Knowledge APPLY SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1477
Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.
EASY
4 Competency UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1428
Find the value of 'a' for which $f(x)=\sqrt{3}\sin x-\cos x-2ax+6$ is decreasing in R.
MEDIUM
5 Knowledge REMEMBER SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1362
The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?
MEDIUM
6 Knowledge REMEMBER SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1254
Find the absolute maximum and absolute minimum values of the function f given by $f(x)=\frac{x}{2}+\frac{2}{x}$ , on the interval [1, 2].
EASY
7 Knowledge APPLY SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1253
Find the intervals in which the function $f(x)=\frac{log~x}{x}$ is strictly increasing or strictly decreasing.
MEDIUM

Long Answer (5 Marks)

1 Competency APPLY LA 5 Mark(s) 2026 AISSCE(Board Exam)
#1816
A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
HARD
2 Competency APPLY LA 5 Mark(s) 2026 AISSCE(Board Exam)
#1815
Find the sub intervals in which $f(x)=\cot^{-1}(\sin x+\cos x)$, $x \in (0,\pi)$ is increasing and decreasing.
HARD
3 Competency UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1416
The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation $y=4x-\frac{1}{2}x^{2}$, where x is the number of days exposed to sunlight. (i) Find the rate of growth of the plant with respect to sunlight. (ii) In how many days will the plant attain its maximum height? What is the maximum height?
MEDIUM
4 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1373
Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.
MEDIUM
5 Competency UNDERSTAND LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1262
The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.
HARD
6 Competency REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1261
It is given that function $f(x)=x^{4}-62x^{2}+ax+9$ attains local maximum value at $x=1$ Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values.
MEDIUM
7 Competency UNDERSTAND LA 5 Mark(s) 2023
#906
(a) The median of an equilateral triangle is increasing at the rate of 2√3 cm/s. Find the rate at which its side is increasing. OR (b) Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
MEDIUM

Subjective / Numerical

1 Competency APPLY SUBJECTIVE 4 Mark(s) 2026-27 CBSE Sample QP
#2099
Water is essential for life, and rivers are vital water sources across the world. The Yamuna Action Plan (YAP), one of India's largest river restoration projects, was launched to clean the Yamuna, but its effectiveness (WQI) the Water Quality Index must be measured through cost-benefit, economic and environmental analysis. Therefore, comprehensive pollutant treatment, prevention of further contamination, and community awareness are crucial for sustainable river management.
Implementing the same concept to the river Yamuna, if an initial base line WQI is 35, a quadratic expression is formed to measure the impact of investment of ₹$x$ crore on the WQI i.e., $W(x)$ as $W(x) = 35 + 1.6x - 0.02x^2$, where $x$ is in Rupees in crores Use the given $W(x)$ to answer the following questions:
(i) Find the value of $x$ at the stationary point for the function $W(x)$. [1 mark]
(ii) Find the maximum value of the WQI using the second derivative test? [1 mark]
(iii) (a) If the investment is ₹ 35 crores, find by how much the WQI is short of its maximum achievable value.[2 marks]
Or
(b) Evaluate the WQI when the investment is ₹60 crore. Determine whether this indicates an enhancement or a diminishing return in our objective when compared with the optimum WQI. [2 marks]
MEDIUM

Case Study & Passage Based

Read the following passage:
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections.

Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be $y$ metres.,
1 Competency APPLY SUBJECTIVE 2 Mark(s) 2025 AISSCE(Board Exam)
#704
Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.
HARD
2 Competency APPLY SUBJECTIVE 1 Mark(s) 2025 AISSCE(Board Exam)
#703
Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
HARD
3 Competency APPLY SUBJECTIVE 1 Mark(s) 2025 AISSCE(Board Exam)
#702
Write the area of the solar panel as a function of $x$
MEDIUM
4 Competency APPLY SUBJECTIVE 1 Mark(s) 2025 AISSCE(Board Exam)
#701
Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of
x and y..
MEDIUM