max min ↗ increasing ↘ decreasing
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Chapter 6  ·  Class XII Mathematics  ·  MCQ Practice

MCQ Practice Arena

Application of Derivatives

Rates, Extrema, and Curves — Turn Calculus into Confident Answers

📋 50 MCQs ⭐ 26 PYQs ⏱ 90 sec/Q
🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs
16Easy
22Medium
12Hard
26PYQs
90 secAvg Time/Q
3Topics
Easy 32% Medium 44% Hard 24%

Why Practise These MCQs?

JEE MainJEE AdvancedCBSE

Maxima-minima and rate-of-change questions are among the most reliably scoring MCQs in both boards and JEE Main, provided the second derivative test is applied cleanly. This bank leans into optimization word problems and increasing/decreasing interval questions, the two forms that appear most often in past papers.

Topic-wise MCQ Breakdown

Rate of Change of Quantities12 Q
Increasing & Decreasing Functions16 Q
Maxima and Minima22 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\dfrac{dV}{dt} = \dfrac{dV}{dr}\cdot\dfrac{dr}{dt}$
$f \text{ is increasing if } f'(x) > 0$
$\text{Critical points: } f'(x) = 0$
$f''(x) < 0 \to \text{maxima}; f''(x) > 0 \to \text{minima}$

MCQ Solving Strategy

In related-rates problems, write the governing relation between the quantities first, then differentiate with respect to time — don't differentiate before you've set up the full equation. For maxima/minima, always confirm a critical point with the second derivative test or a sign change of f', and in closed-interval problems always check the endpoints too.

⚠ Common Traps & Errors

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Basic rate-of-change substitution, identify critical points

② Medium

Increasing/decreasing interval MCQs, first derivative test applications

③ Hard

Optimization word problems, second derivative test with constraints

★ PYQ

JEE Main — optimization with a constraint equation; CBSE — monotonicity proofs

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 🎯 PYQ Bank Previous year questions ✔️❌ True / False Concept-based True & False questions 📘✔️ Exercises Step by step Solutions to textbook exercises ➡ Next Chapter Integrals MCQs
🎯 Knowledge Check

Mathematics — APPLICATION OF DERIVATIVES

50 Questions Class 12 MCQs
1
If \(y=x^2\), then \(\dfrac{dy}{dx}\) at \(x=3\) is:
2
A function \(f(x)\) is increasing on an interval if:
3
The function \(f(x)=x^2\) is increasing on:
4
The function \(f(x)=x^2\) is decreasing on:
5
If \(s=t^2+3t\) represents displacement, then the velocity at \(t=2\) is:
6
The rate of change of the area \(A=\pi r^2\) with respect to radius \(r\) is:
7
For \(f(x)=x^3-3x\), the critical points are:
8
The function \(f(x)=x^3-3x\) is increasing on:
9
The function \(f(x)=x^3-3x\) is decreasing on:
10
If \(f'(x)>0\) for every \(x\) in an interval \(I\), then \(f\) is:
11
The function \(f(x)=x^2-4x+5\) has its minimum value at:
12
The minimum value of \(f(x)=x^2-4x+5\) is:
13
The maximum value of \(f(x)=-x^2+4x+1\) occurs at:
14
The maximum value of \(f(x)=-x^2+4x+1\) is:
15
If \(f'(x)\) changes from negative to positive at \(x=a\), then \(f(a)\) is:
16
If \(f'(x)\) changes from positive to negative at \(x=a\), then \(f(a)\) is:
17
The radius of a circle is increasing at \(2\) cm/s. The rate at which its area increases when \(r=5\) cm is:
18
The side of a square increases at \(3\) cm/s. The rate of increase of its area when the side is \(4\) cm is:
19
The volume of a sphere is \(V=\frac{4}{3}\pi r^3\). If \(\dfrac{dr}{dt}=2\) cm/s, then \(\dfrac{dV}{dt}\) at \(r=3\) cm is:
20
The length of a rectangle is increasing at \(2\) cm/s and its breadth is increasing at \(3\) cm/s. If \(l=5\) cm and \(b=4\) cm, the rate of increase of its area is:
21
The function \(f(x)=x^3-6x^2+9x+1\) has a local maximum at:
22
For \(f(x)=x^3-6x^2+9x+1\), the local minimum value is:
23
The function \(f(x)=x+\frac{1}{x}\), \(x>0\), has a minimum at:
24
The minimum value of \(x+\frac{1}{x}\), for \(x>0\), is:
25
The function \(f(x)=x^2+\frac{4}{x}\), \(x>0\), has a critical point at:
26
The function \(f(x)=x^2+\frac{4}{x}\), \(x>0\), attains its minimum value at:
27
The equation of the tangent to \(y=x^2\) at \(x=1\) is:
28
The equation of the normal to \(y=x^2\) at \(x=1\) is:
29
The slope of the tangent to \(y=x^3-3x+2\) at \(x=1\) is:
30
A function \(f(x)\) is increasing on an interval when:
31
The function \(f(x)=x^3-12x+5\) is increasing on:
32
The function \(f(x)=x^3-12x+5\) has a local maximum at:
33
The local maximum value of \(f(x)=x^3-12x+5\) is:
34
If the radius of a spherical balloon increases at \(0.5\) cm/s, the rate of increase of its volume at \(r=4\) cm is:
35
A particle moves according to \(s=t^3-6t^2+9t\). Its velocity is zero at:
36
For \(s=t^3-6t^2+9t\), the particle is at rest and changes its direction at:
37
The greatest value of \(x(6-x)\) for \(x\in[0,6]\) is:
38
Two positive numbers have sum \(20\). Their product is maximum when the numbers are:
39
Among all rectangles with perimeter \(40\) cm, the maximum area is:
40
Among all rectangles with fixed area \(A\), the rectangle having the least perimeter is:
41
A box with square base has volume \(500\) cm\(^3\). If its base side is \(x\) cm and height is \(h\) cm, then \(h\) equals:
42
An open-top box with square base of side \(x\) and volume \(V\) has surface area:
43
For an open-top box of fixed volume \(V\) with square base side \(x\), the surface area is minimum when:
44
The point on the parabola \(y^2=4x\) nearest to the point \((4,0)\) is:
45
The maximum value of \(x^2(4-x)\), for \(0\leq x\leq4\), is:
46
For the function \(f(x)=\frac{x}{1+x^2}\), the maximum value for \(x>0\) is:
47
The minimum value of \(x+\frac{9}{x}\), \(x>0\), is:
48
A cone has fixed volume \(V\). If its radius is \(r\) and height is \(h\), then \(h\) in terms of \(r\) is:
49
A rectangular sheet has perimeter \(36\) cm. The dimensions that give maximum area are:
50
A right circular cylinder has fixed volume \(V\). Its total surface area is minimum when the relation between its height \(h\) and radius \(r\) is:
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NCERT Class 12 Application of Derivatives: 50 MCQs
NCERT Class 12 Application of Derivatives: 50 MCQs — Complete Notes & Solutions · academia-aeternum.com
NCERT Class 12 Mathematics Chapter 6: Application of Derivatives MCQs Prepare effectively for NCERT Class 12 Mathematics Chapter 6 – Application of Derivatives with this carefully structured collection of 50 multiple-choice questions (MCQs). These questions cover important concepts such as increasing and decreasing functions, critical points, local maxima and minima, rate of change, tangents and normals, approximation, and practical problems based on maxima and minima. The MCQs are arranged in…
🎓 Class 12 📐 Mathematics 📖 NCERT ✅ Free Access 🏆 CBSE · JEE
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Frequently Asked Questions

Application of Derivatives is NCERT Class 12 Mathematics Chapter 6, covering rate of change, increasing and decreasing functions, maxima and minima, and optimization problems.

The rate of change of y with respect to x is given by dy/dx. It measures how rapidly one quantity changes with respect to another.

A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0 on the given interval.

Critical points are points in the domain where f'(x) = 0 or where f'(x) does not exist, provided the function is defined at those points.

If f'(x) changes from positive to negative at a critical point, the function has a local maximum. If it changes from negative to positive, the function has a local minimum.

If f'(c) = 0 and f''(c) < 0, the function has a local maximum at c. If f''(c) > 0, it has a local minimum at c. If f''(c) = 0, the test is inconclusive.

Find all critical points in the interior of the interval, include both endpoints, evaluate the function at all candidate points, and compare the values.

Related rates problems involve quantities that change with time and are connected by a mathematical relation. Differentiating that relation with respect to time gives the required rate.

Derivatives help identify values that maximize or minimize quantities such as area, volume, distance, cost, revenue and profit, subject to given conditions.

Application of Derivatives provides important calculus concepts used in optimization, monotonicity, maxima-minima, related rates and other mathematical problems in competitive entrance examinations.

Application of Derivatives covers increasing and decreasing functions, rate of change, tangents and normals, approximation, maxima and minima, and optimization problems.

True and False questions help students test their understanding of derivative-based concepts, identify misconceptions, and strengthen conceptual accuracy.

If \(f'(x)>0\) throughout an interval, then \(f(x)\) is strictly increasing on that interval.

If \(f'(x)<0\) throughout an interval, then \(f(x)\) is strictly decreasing on that interval.

A critical point is a point in the domain where \(f'(x)=0\) or where \(f'(x)\) does not exist, provided the function itself is defined there.

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