If the rate of change of volume of a sphere is twice the rate of change of its radius, then the surface area of the sphere is :
(A) $1 \text{ sq unit}$
(B) $2 \text{ sq units}$
(C) $3 \text{ sq units}$
(D) $4 \text{ sq units}$
CBSE CLASS 12 • MATHEMATICS • CHAPTER 06
Applications of Derivatives
64 distinct questions across 81 board paper set appearances (2022–2026).
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
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 $(-\infty, 0)$
If M and m denote the local maximum and local minimum values of the function $f(x) = x + \frac{1}{x} (x \neq 0)$ respectively, find the value of $(M - m)$.
(a) 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.
OR
(b) The perimeter of a rectangular metallic sheet is $300 \text...
A housing society wants to commission a swimming pool for its residents. For this, they have to purchase a square piece of land and dig this to such a depth that its capacity is 250 cubic metres. Cost of land is ₹ 500 per square metre. The cost of digging increases with the depth and cost for the wh...
The rate of change of volume of a sphere with respect to its diameter, when its radius is $5 \text{ cm}$, is :
(A) $400\pi \text{ cm}^3/\text{cm}$
(B) $100\pi \text{ cm}^3/\text{cm}$
(C) $50\pi \text{ cm}^3/\text{cm}$
(D) $25\pi \text{ cm}^3/\text{cm}$
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of $300\text{ m}$ and even function in the dark.
A camera is installed on a pole at the height of $5\text{ m}$. It detects a car...
A ladder of fixed length 'h' is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions : (i) Express the distance (y) between the wall and foot of the ladder in terms of 'h' and height (x) on the wall at...
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text...
(i) What are the critical points of the function f(x)?
For $f(x) = x + \frac{1}{x}$ ($x \neq 0$)
(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
For $f(x) = x + \frac{1}{x}$ ($x \neq 0$)
(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
For $f(x) = x + \frac{1}{x} \text{ } (x \neq 0)$
(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
22. (a) The volume of a spherical balloon is increasing at the rate of $20\text{ cm}^3\text{/s}$. Find the rate of change of its surface area when its radius is $4\text{ cm}$.
OR
(b) Find the intervals on which the function $f(x) = x^3 + 2x^2 - 1$ is strictly increasing.
The rate of change of volume of a sphere with respect to its diameter, when its radius is $5$ cm, is :
(A) $400\pi \text{ cm}^3/\text{cm}$
(B) $100\pi \text{ cm}^3/\text{cm}$
(C) $50\pi \text{ cm}^3/\text{cm}$
(D) $25\pi \text{ cm}^3/\text{cm}$
A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.
The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.
The values of $x$ for which the rate of increase of $f(x) = x^3 - 5x^2 + 6x - 7$ is thrice the rate of increase of $x$, are :
(A) $-3, -\frac{1}{3}$
(B) $3, -\frac{1}{3}$
(C) $3, \frac{1}{3}$
(D) $-3, \frac{1}{3}$
Find the values of '$a$' and '$b$', if $y = \frac{ax - b}{(x - 1)(x - 4)}$ has a critical point $(3, 1)$.
If the rate of change of the area of a circle is equal to the rate of change of its diameter, then its radius is :
(A) $2\pi \text{ units}$
(B) $\pi \text{ units}$
(C) $\frac{\pi}{2} \text{ units}$
(D) $\frac{1}{\pi} \text{ units}$
Case Study -2
LED bulbs are energy-efficient because they use significantly less electricity than traditional bulbs while producing the same amount of light. They convert more energy into light rather than heat, reducing waste. Additionally, their long lifespan means fewer replacements, saving reso...
(a) Prove that the function $f(x) = x^2 - x + 1$ is neither strictly increasing nor strictly decreasing on the interval $(-1, 1)$.
OR
(b) Find $\frac{dy}{dx}$, if $y^x + x^y + x^x = a^b$.
The radius of a cylinder is increasing at the rate of $3\text{ cm/s}$, and its height is decreasing at the rate of $5\text{ cm/s}$. Find the rate of change of its volume, when radius is $4\text{ cm}$ and height is $7\text{ cm}$.
Find the intervals in which the function $f(x) = \frac{3}{2}x^4 - 4x^3 - 45x^2 + 51$ is :
(a) strictly increasing.
(b) strictly decreasing.
(a) If $y = \cos x^2 + \cos^2 x + \cos^2 (x^2) + \cos (x^x)$, find $\frac{\text{dy}}{\text{dx}}$.
OR
(b) Find the intervals in which the function given by $f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11$ is :
(i) strictly increasing.
(ii) strictly decreasing.
(a) If $y = \cos x^2 + \cos^2 x + \cos^2 \left(x^2\right) + \cos \left(x^x\right)$, find $\frac{\text{d}y}{\text{d}x}$.
OR
(b) Find the intervals in which the function given by
$f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11$ is :
(i) strictly increasing.
(ii) strictly decreasin...
A spherical balloon has a variable diameter $\frac{2}{3}(2\text{t} + 1)$. Find the rate of change of its volume with respect to t.
Find the intervals in which the function f given by $\text{f(x)} = -2\text{x}^3 - 9\text{x}^2 - 12\text{x} + 1$ is :\n(i) strictly increasing.\n(ii) strictly decreasing.
(b) Find the intervals in which the function given by $f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11$ is :
(i) strictly increasing.
(ii) strictly decreasing.
Show that the function $f(x) = 4x^3 - 18x^2 + 27x - 7$ has neither maxima nor minima.
(iii) (a) 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.
A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is proportional to its surface area. Calculate the rate of decrease of its radius.
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 $x$ m...
Find the intervals in which the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is (a) strictly increasing (b) strictly decreasing
Find the local maxima and local minima of the function $f(x) = \frac{8}{3}x^3 - 12x^2 + 18x + 5$.
The radius of a cylinder is decreasing at a rate of $2\text{ cm/s}$ and the altitude is increasing at the rate of $3\text{ cm/s}$. Find the rate of change of volume of this cylinder when its radius is $4\text{ cm}$ and altitude is $6\text{ cm}$.
Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of $2\text{ cm/s}$, find the rate of increase of the outer radius when the radii are $2\text{ cm}$ and $4\text{ cm}$ respectively.
Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point.
Show that the area (A) of the right triangle is maximum at the critical point.
OR If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall ?
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) $125\pi$
Find local maximum value and local minimum value (whichever exists) for the function $f(x) = 4x^2 + \frac{1}{x} \; (x \neq 0)$.
According to a psychologist, the ability of a person to understand spatial concepts is given by $A = \frac{1}{3}\sqrt{t}$, where $t$ is the age in years, $t \in [5, 18]$. Show that the rate of increase of the ability to understand spatial concepts decreases with age in between $5$ and $18$.
Check whether V has a point of inflection at $x = \frac{65}{6}$ or not?
Assertion (A) : If the side of a square is increasing at the rate of $0\cdot 2$ cm/s, then the rate of increase of its perimeter is $0\cdot 8$ cm/s.
Reason (R) : Perimeter of a square $= 4$ (side).
Find the intervals on which the function $f(x) = 10 - 6x - 2x^2$ is (a) strictly increasing (b) strictly decreasing.
If the sides of a square are decreasing at the rate of $1.5 \text{ cm/s}$, the rate of decrease of its perimeter is :
(A) $1.5 \text{ cm/s}$
(B) $6 \text{ cm/s}$
(C) $3 \text{ cm/s}$
(D) $2.25 \text{ cm/s}$
The rate of change of surface area of a sphere with respect to its radius 'r', when $r = 4 \text{ cm}$, is :
(A) $64\pi \text{ cm}^2/\text{cm}$
(B) $48\pi \text{ cm}^2/\text{cm}$
(C) $32\pi \text{ cm}^2/\text{cm}$
(D) $16\pi \text{ cm}^2/\text{cm}$
Find the intervals in which the function $f(x) = \frac{\log x}{x}$ is strictly increasing or strictly decreasing.
The surface area of a cube increases at the rate of $72 \text{ cm}^2/\text{sec}$. Find the rate of change of its volume, when the edge of the cube measures $3 \text{ cm}$.
The surface area of a cube increases at the rate of $72 \ \text{cm}^2/\text{sec}$. Find the rate of change of its volume, when the edge of the cube measures $3 \ \text{cm}$.
(ii) Find the critical point(s) of the area function A(x).
For the function $f(x) = x^3$, $x = 0$ is a point of : (A) local maxima (B) local minima (C) non-differentiability (D) inflexion
If the radius of a circle is increasing at the rate of $0.5 \text{ cm/s}$, then the rate of increase of its circumference is : (A) $\frac{2\pi}{3} \text{ cm/s}$ (B) $\pi \text{ cm/s}$ (C) $\frac{4\pi}{3} \text{ cm/s}$ (D) $2\pi \text{ cm/s}$
If the radius of a circle is increasing at the rate of 0.5 cm/s, then the rate of increase of its circumference is :
(A) $\frac{2\pi}{3}$ cm/s
(B) $\pi$ cm/s
(C) $\frac{4\pi}{3}$ cm/s
(D) $2\pi$ cm/s
(ii) Find rate of change of radius at an instant when $r = 2\sqrt{2}\text{ cm}$.
(iii) (b) Find the rate of change of height 'h' at an instant when slant height is $4\text{ cm}$.
If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.
(a) Determine for what values of x, the function $f(x) = x^4 - \frac{x^3}{3}$ is strictly increasing or strictly decreasing. OR (b) A firm has the following total cost and demand functions : $C(x) = \frac{x^3}{3} - 7x^2 + 111x + 50$ and $x = 100 - p$ (i) Find the total revenue function in terms of x...
Find the interval in which the function $x^3 - 12x^2 + 36x + 17$ is strictly increasing.
The rate of change of the volume of a spherical bubble with respect to its radius r at $r = 3 \text{ cm}$ is :
(a) $24 \pi \text{ cm}^3/\text{cm}$
(b) $36 \pi \text{ cm}^2/\text{cm}$
(c) $36 \pi \text{ cm}^3/\text{cm}$
(d) $24 \pi \text{ cm}^2/\text{cm}$
(iii) Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
(iii) Find the points of local maximum/local minimum, if any, in the interval $(0, 12)$ as well as the points of absolute maximum/absolute minimum in the interval $[0, 12]$. Also, find the corresponding loc...
Case-Study 2: Read the following passage and answer the questions given below.
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
(i) If the length and...