Application of Derivatives

CBSE Class 12 Mathematics · every board question from this chapter, 2023–2026

• A cylindrical tank is being filled with sand at a rate of 314 m^3 / h. If the radius of the tank is 10 m, then the height of sand…2026
• The surface area of a sphere when its volume changes at the same rate as its radius is:2026
• If the distance travelled by a particle in t seconds is given by S=72 t+3 t^2-t^3, then time taken by the particle to come to…2026
• The least value of f(x)=x^3-12 x, x ∈[0,3] is2026
• Absolute minimum value of f(x)=(x-2)^2+5 in the interval [-3,2] is:2026
• The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is:2026
• The least value of f(x)=e^-x in [0, 3] is2026
• The absolute maximum value of f(x)=x^2+1 in [-5,2] is2026
• Find the sub-interval(s) of (0, (π)/2) in which f(x)= tan x-4 x is increasing.2026
• A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume…2026
• Find the interval(s) for which the function f(x)=x/3+3/x, x ≠ 0 is increasing.2026
• Find the interval(s) in which the function f(x)=x/(log x), where x ∈(0,1) ∪(1, ∞), is increasing.2026
• Find the sub-interval of (0, ∞) in which f(x)=x log x is increasing.2026
• Find the absolute maximum value of f(x)= cos x+ sin^2 x, x ∈[0, π] OR If the volume of a solid hemisphere increases at a uniform…2026
• Find the values of x for which f(x)=x^x, x>0 is increasing.2026
• Determine the values of x for which f(x)=(x-3)/(x+1), x ≠-1 is an increasing function.2026
• Determine the interval(s) in which f(x)=x/3+3/x, x ≠ 0 is increasing.2026
• Determine the interval(s) in which f(x)=5 x^3 / 2-3 x^5 / 2, x>0 is increasing.2026
• Find the sub-interval of (0, ∞) in which f(x)=x^2 e^-x is increasing.2026
• A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that…2026
• 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…2026
• 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…2026
• Find the sub-interval of (0, (π)/2) in which f(x)= log ( sin x+ cos x) is increasing and decreasing. OR A rectangle of perimeter…2026
• Find the sub intervals in which f(x)= cot^-1( sin x+ cos x), x ∈(0, π) is increasing and decreasing. OR A rectangle of perimeter…2026
• Find the sub-interval of (0, π) in which f(x)= tan^-1( sin x- cos x) is increasing and decreasing. OR A rectangle of perimeter 24…2026
• At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and…2026
• A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area…2026
• An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for…2026
• Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure.…2026
• The function f(x)=x^2-4 x+6 is increasing in the interval2025
• The values of λ so that f(x)= sin x- cos x-λ x+C decreases for all real values of x are:2025
• The absolute maximum value of function f(x)=x^3-3 x+2 in [0,2] is:2025
• f(x)=x^x has a critical point at:2025
• The slope of the curve y=-x^3+3 x^2+8 x-20 is maximum at:2025
• Edge of a variable cube increases at the rate of 5 cm / s. The rate at which the surface area of the cube increases when the edge…2025
• A spherical ball has a variable diameter 5/2(3 x+1). The rate of change of its volume w.r.t. x, when x=1, is:2025
• A cylindrical tank of radius 10 cm is being filled with sugar at the rate of 100 π cm^3 / s. The rate, at which the height of the…2025
• Find the values of 'a' for which f(x)=x^2-2 a x+b is an increasing function for x>0.2025
• Find the least value of ' a ' so that f(x)=2 x^2-a x+3 is an increasing function on [2,4]. OR If f(x)=x+1/x, x ≥ 1, show that f…2025
• Find the values of ' a ' for which f(x)=√3 sin x- cos x-2 a x+b is decreasing on R.2025
• Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of 2 cm / s, find the rate of…2025
• The radius of a cylinder is decreasing at a rate of 2 cm / s and the altitude is increasing at the rate of 3 cm / s. Find the…2025
• Surface area of a balloon (spherical), when air is blown into it, increases at a rate of 5 mm^2 / s. When the radius of the…2025
• Determine the values of x for which f(x)=(x-4)/(x+1), x ≠-1 is an increasing or a decreasing function.2025
• Find the interval in which f(x)=x+1/x is always increasing, x ≠ 0.2025
• Find the values of' a 'for which f(x)= sin x-a x+b is increasing on R.2025
• Determine those values of x for which f(x)=2/x-5, x ≠ 0 is increasing or decreasing.2025
• Find the intervals in which function f(x)=5 x^(3/2)-3 x^(5/2) is (i) increasing (ii) decreasing.2025
• Find the local maxima and local minima of the function f(x)=8/3 x^3-12 x^2+18 x+5.2025
• For the curve y=5 x-2 x^3, if x increases at the rate of 2 units / s, then how fast is the slope of the curve changing when x=2?2025
• A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm^3 / s from the leak.…2025
• 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…2025
• Show that f(x)= tan^-1( sin x+ cos x) is an increasing function in [0, (π)/4].2025
• Find the interval/intervals in which the function f(x)= sin 3 x- cos 3 x, 0<x<(π)/2 is strictly increasing.2025
• The area of an expanding rectangle is increasing at the rate of 48 cm^2 / s. The length of the rectangle is always square of its…2025
• Show that of all the rectangles with a fixed perimeter, the square has the greatest area.2025
• Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.2025
• Find dimensions of a rectangle of perimeter 12 cm which will generate maximum volume when swept along a circular rotation keeping…2025
• Find the absolute maximum and absolute minimum of function f(x)=2 x^3-15 x^2+36 x+1 on [1,5].2025
• The relation between the height of the plant ( ycm ) with respect to exposure to sunlight is governed by the equation y=4 x-1/2…2025
• A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is…2025
• A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The…2025
• 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…2025
• A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The…2025
• A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity…2025
• If the sides of a square are decreasing at the rate of 1.5 cm / s, the rate of decrease of its perimeter is:2024
• The function f(x)=x^3-3 x^2+12 x-18 is:2024
• The function f(x)=x/2+2/x has a local minima at x equal to:2024
• Given a curve y=7 x-x^3 and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing,…2024
• The rate of change of surface area of a sphere with respect to its radius ' r ', when r=4 cm, is:2024
• The point of inflexion of a function f(x) is the point where2024
• 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,…2024
• The function f(x)=k x- sin x is strictly increasing for2024
• Determine whether the function f(x)=x^2-6 x+3 is increasing or decreasing in [4,6].2024
• Show that the function f(x)=4 x^3-18 x^2+27 x-7 has neither maxima nor minima.2024
• Show that the function f given by f(x)= sin x+ cos x, is strictly decreasing in the interval ((π)/4, (5 π)/4).2024
• Find the interval in which the function f(x)=x^4-4 x^3+10 is strictly decreasing.2024
• Show that f(x)=(4 sin x)/(2+ cos x)-x is an increasing function of x in [0, (π)/2].2024
• Sand is pouring from a pipe at the rate of 15 cm^3 / minute. The falling sand forms a cone on the ground such that the height of…2024
• 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…2024
• If M and m denote the local maximum and local minimum values of the function f(x)=x+1/x(x ≠ 0) respectively, find the value of…2024
• Find local maximum value and local minimum value (whichever exists) for the function f(x)=4 x^2+1/(x)(x ≠ 0).2024
• The surface area of a cube increases at the rate of 72 cm^2 / sec. Find the rate of change of its volume, when the edge of the…2024
• 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…2024
• Find the absolute maximum and minimum values of the function f(x)=12 x^4 / 3-6 x^1 / 3, x ∈[0,1].2024
• Show that f(x)=e^x-e^-x+x- tan^-1 x is strictly increasing in its domain.2024
• Find the intervals in which the function f(x)=(log x)/x is strictly increasing or strictly decreasing. OR Find the absolute…2024
• It is given that function f(x)=x^4-62 x^2+a x+9 attains local maximum value at x=1. Find the value of ' a ', hence obtain all…2024
• Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While…2024
• A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1…2024
• The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can…2024
• A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator…2024
• The interval in which the function f(x)=2 x^3+9 x^2+12 x-1 is decreasing, is2023
• The function f(x)=x^3+3 x is increasing in interval2023
• If f(x)=a(x- cos x) is strictly decreasing in R, then ' a ' belongs to2023
• If the product of two positive numbers is 9, find the numbers so that the sum of their squares is minimum.2023
• Consider the statement "There exists at least one value of b ∈ R for which f(x)=b/x, b ≠ 0 is strictly increasing in R-{0}."…2023
• Find the interval in which the function f(x)=2 x^3-3 x is strictly increasing.2023
• Find the maximum and minimum values of the function given by f(x)=5+ sin 2 x.2023
• Find the sub-intervals in which f(x)= log (2+x)-x/(2+x), x>-2 is increasing or decreasing.2023
• If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm / sec, how fast is the area of…2023
• If f(x)=a( tan x- cot x), where a>0, then find whether f(x) is increasing or decreasing function in its domain.2023
• Find the points on the curve 6 y=x^3+2 at which ordinate is changing 8 times as fast as abscissa.2023
• Find the point on the curve y^2=8 x for which the abscissa and ordinate change at the same rate.2023
• Show that the function f(x)=(16 sin x)/(4+ cos x)-x, is strictly decreasing in ((π)/2, π).2023
• If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly…2023
• 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…2023
• A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to…2023
• In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base…2023
• Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside…2023
• Sooraj's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the…2023
• A volleyball player serves the ball which takes a parabolic path given by the equation h(t)=-7/2 t^2+13/2 t+1, where h(t) is the…2023
• The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and…2023

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