Are these practice questions for Application of Derivatives based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Application of Derivatives free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Application of Derivatives mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Application of Derivatives for full coverage.

How many practice questions are there for Application of Derivatives?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Application of Derivatives Class 12 Mathematics Practice Questions with Answers

easy

Justify why the function f(x) = \log(x) has no local maxima or minima on its domain (0, \infty).

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easy

Identify the term used for a point c in the domain of a function f where either f'(c) = 0 or f is not differentiable at c.

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easy

State the condition, using the first derivative, for a function f(x) to be strictly decreasing on an open interval (a, b).

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easy

Explain the relationship between the sign of the first derivative of a function,

f'(x)

, and the increasing or decreasing nature of the function

f(x)

.

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easy

Justify the statement: The rate of change of the volume of a sphere with respect to its radius is numerically equal to its surface area.

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easy

Analyze if the function

f(x) = 1 - 6x - 3x^2

is increasing or decreasing at

x = -2

.

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easy

The side of a square is increasing at the rate of

0.2 \text{ cm/s}

. Calculate the rate of increase of its perimeter.

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easy

Define the term 'marginal cost' in the context of economics.

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easy

The total cost function for producing

x

units is

C(x) = 100 + 50x - 0.5x^2

. Calculate the marginal cost when 10 units are produced.

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easy

Evaluate the approximate change in the volume of a sphere when its radius increases from 10 cm to 10.02 cm. Propose a formula for this approximation using differentials.

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medium

Explain the difference between a 'local minimum' and an 'absolute minimum' for a function defined on a closed interval

[a, b]

.

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medium

Explain the concept of 'rate of change of quantities' using derivatives. Illustrate with the example of a melting snowball (a sphere), explaining how the rate of change of its surface area is related to the rate of change of its radius.

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medium

A student finds that for f(x) = (x-2)^4, f'(2) = 0 and f''(2) = 0. They conclude the second derivative test fails and stop. Critique this approach and propose the necessary next step.

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medium

A function

f(x)

has a critical point at

x=c

. If

f'(c) = 0

and

f''(c) = -5

, examine if

x=c

is a point of local maximum or local minimum.

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medium

Describe the First Derivative Test for determining if a critical point

c

is a point of local minima.

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medium

Recall the formula for the rate of change of y with respect to x, given that two variables x and y are varying with respect to a third variable t.

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medium

A student claims that if f'(c) = 0 and f''(c) > 0, then c is a point of absolute minima for the function f on any closed interval [a, b] containing c. Critique this claim.

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medium

Find the critical points of the function

f(x) = x^3 - 12x + 5

.

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medium

Two cars start from the same point at the same time. Car A travels north at 60 km/h and Car B travels east at 80 km/h. Formulate an expression for the rate at which the distance between the cars is increasing after 2 hours and evaluate it.

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medium

A particle moves along the curve y = x^3 + 2x. At what point(s) on the curve is the y-coordinate changing twice as fast as the x-coordinate? Justify your answer.

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medium

Prove that the rectangle of maximum area that can be inscribed in a circle of radius R is a square. Justify each step of the derivation, including setting up the function and using a derivative test to confirm the maximum.

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medium

Prove that for x > 0, the function f(x) = x + \frac{1}{x} has a local minimum. Then, justify that this local minimum value is 2.

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medium

Summarize the steps required to find the absolute maximum and absolute minimum values of a continuous function

f(x)

on a closed interval

[a, b]

.

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medium

Find the local maximum and local minimum values of the function

f(x) = x^3 - 6x^2 + 9x + 1

.

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medium

Calculate the equation of the tangent to the curve

y = x^3 - x + 1

at the point where

x=1

.

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medium

The radius of a spherical soap bubble is decreasing at the rate of

0.5 \text{ cm/s}

. At what rate is its volume decreasing when the radius is

2 \text{ cm}

?

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medium

Find the absolute maximum and minimum values of the function

f(x) = x^3 - 3x^2 + 5

on the interval

[-1, 3]

.

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medium

The total revenue

R(x)

in Rupees from the sale of

x

items is given by

R(x) = 10x^2 + 20x + 5

. Explain how to find the marginal revenue and describe what this value represents.

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medium

Find the intervals in which the function

f(x) = 2x^3 - 9x^2 + 12x + 15

is strictly increasing or decreasing.

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medium

Summarize the complete procedure for finding and classifying all local extrema (local maxima and local minima) for a given differentiable function

f(x)

. Your summary should detail how to find critical points and how to use both the First and Second Derivative Tests.

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medium

According to the Second Derivative Test, what are the two conditions that must be met for a point

x = c

to be a point of local maxima for a function

f

?

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hard

Formulate a condition using the signs of the first derivative, f'(x), to identify a point of inflection c where f'(c) = 0.

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hard

Show that the altitude of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height

H

and radius

R

is one-third the height of the cone.

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hard

Demonstrate that the function

f(x) = \log(\sin x)

is strictly increasing on the interval

(0, \frac{\pi}{2})

.

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hard

Justify the intervals where the function f(x) = \sin(x) - \cos(x) is strictly increasing or decreasing on the interval (0, 2\pi).

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hard

Sand is being poured from a pipe at the rate of

16 \text{ cm}^3/\text{s}

. The falling sand forms a cone on the ground in such a way that the diameter of the base is always equal to its height. How fast is the height of the sand cone increasing when the height is

4 \text{ cm}

?

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hard

Describe a scenario in which the Second Derivative Test is inconclusive or fails, and explain the subsequent procedure to classify the critical point.

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hard

A window is designed in the form of a rectangle surmounted by a semicircle. The total perimeter of the window is 10 m. Formulate an expression for the area of the window as a function of the radius of the semicircle. Propose the dimensions of the window that will admit the maximum possible light and justify your proposal.

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hard

Design a container in the shape of a right circular cylinder with an open top that has a fixed surface area of 100\pi cm². Formulate the volume V as a function of the radius r. Determine the dimensions (radius and height) that will maximize the volume, and justify your result using the second derivative test.

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hard

If a function

f(x)

has a derivative

f'(x) = x^2

, explain whether the point

x = 0

is a local maximum, local minimum, or neither, using the First Derivative Test.

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hard

A wire of length 20 m is to be cut into two pieces. One piece will be bent to form a square and the other to form an equilateral triangle. What should be the lengths of the two pieces so that the combined area of the two is minimum?

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hard

Describe the concept of a point of inflection. Explain its relationship to the first and second derivatives of a function. How does the First Derivative Test identify a point of inflection?

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hard

An airplane is flying at a constant altitude of 4 km and a constant speed of 720 km/h. It passes directly over a radar station. Formulate an equation relating the horizontal distance of the plane from the station and the direct distance between them. Determine the rate at which the distance from the plane to the station is increasing when it is 5 km away from the station.

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hard

A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s. Formulate a method to find how fast the angle between the ladder and the ground is decreasing when the foot of the ladder is 5 m away from the wall.

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Application of DerivativesClass 12 Mathematics Practice Questions