myCBSEguide
- Mathematics
- Class 12 Maths Case...
Class 12 Maths Case Study Questions
Table of Contents
myCBSEguide App
Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.
Class 12 Maths question paper will have 1-2 Case Study Questions. These questions will carry 5 MCQs and students will attempt any four of them. As all of these are only MCQs, it is easy to score good marks with a little practice. Class 12 Maths Case Study Questions are available on the myCBSEguide App and Student Dashboard .
Why Case Studies in CBSE Syllabus?
CBSE has introduced case study questions in the CBSE curriculum recently. The purpose was to make students ready to face real-life challenges with the knowledge acquired in their classrooms. It means, there was a need to connect theories with practicals. Whatsoever the students are learning, they must know how to apply it in their day-to-day life. That’s why CBSE is emphasizing case studies and competency-based education .
Case Study Questions in Maths
Let’s have a look over the class 12 Mathematics sample question paper issued by CBSE, New Delhi. Question numbers 17 and 18 are case study questions.
Focus on concepts
If you go through each MCQ there, you will find that the theme/case study is common but the questions are based on different concepts related to the theme. It means, that if you have done ample practice on the various concepts, you can solve all these MCQs in minutes.
Easy Questions with a Practical Approach
The difficulty level of the questions is average or say easy in some cases. On the other hand, you get four options to choose from. So, you get two levels of support to get full marks with very little effort.
Practice Questions Regularly
Most of the time we feel that it’s easy and neglect it. But in the end, we have to pay for this negligence. This may happen here too. Although it’s easy to score good marks on the case study questions if you don’t practice such questions, you may lose your marks. So, we suggest students should practice at least 30-40 such questions before writing the board exam.
12 Maths Case-Based Questions
We are giving you some examples of case study questions here. We have arranged hundreds of such questions chapter-wise on the myCBSEguide App. It is the complete guide for CBSE students. You can download the myCBSEguide App and get more case study questions there.
Case Study Question – 1
- A is a diagonal matrix
- A is a scalar matrix
- A is a zero matrix
- A is a square matrix
- If A and B are two matrices such that AB = B and BA = A, then B 2 is equal to
Case Study Question – 2
- 4(x 3 – 24x 2 + 144x)
- 4(x 3 – 34x 2 + 244x)
- x 3 – 24x 2 + 144x
- 4x 3 – 24x 2 + 144x
- Local maxima at x = c 1
- Local minima at x = c 1
- Neither maxima nor minima at x = c 1
- None of these
Case Study Questions Matrices -1
Answer Key:
Case Study Questions Matrices – 2
Read the case study carefully and answer any four out of the following questions: Once a mathematics teacher drew a triangle ABC on the blackboard. Now he asked Jose,” If I increase AB by 11 cm and decrease the side BC by 11 cm, then what type of triangle it would be?” Jose said, “It will become an equilateral triangle.”
Again teacher asked Suraj,” If I multiply the side AB by 4 then what will be the relation of this with side AC?” Suraj said it will be 10 cm more than the three times AC.
Find the sides of the triangle using the matrix method and answer the following questions:
- (a) 3 × 3
Case Study Questions Determinants – 01
DETERMINANTS: A determinant is a square array of numbers (written within a pair of vertical lines) that represents a certain sum of products. We can solve a system of equations using determinants, but it becomes very tedious for large systems. We will only do 2 × 2 and 3 × 3 systems using determinants. Using the properties of determinants solve the problem given below and answer the questions that follow:
Three shopkeepers Ram Lal, Shyam Lal, and Ghansham are using polythene bags, handmade bags (prepared by prisoners), and newspaper envelopes as carrying bags. It is found that the shopkeepers Ram Lal, Shyam Lal, and Ghansham are using (20,30,40), (30,40,20), and (40,20,30) polythene bags, handmade bags, and newspapers envelopes respectively. The shopkeeper’s Ram Lal, Shyam Lal, and Ghansham spent ₹250, ₹270, and ₹200 on these carry bags respectively.
- (b) Shyam Lal
- (a) Ram Lal
Case Study Questions Determinants – 02
Case study questions application of derivatives.
- R(x) = -x 2 + 200x + 150000
- R(x) = x 2 – 200x – 140000
- R(x) = 200x 2 + x + 150000
- R(x) = -x 2 + 100 x + 100000
- R'(x) > 0
- R'(x) < 0
- R”(x) = 0
- (a) -x 2 + 200x + 150000
- (a) R'(x) = 0
- (c) 257, -63
Case Study Questions Vector Algebra
- tan−1(5/12)
- tan−1(12/3)
- (b) 130 m/s
- (a) tan−1(5/12)
- (b) 170 m/s
More Case Study Questions
These are only some samples. If you wish to get more case study questions for CBSE class 12 maths, install the myCBSEguide App. It has class 12 Maths chapter-wise case studies with solutions.
12 Maths Exam pattern
Question Paper Design of CBSE class 12 maths is as below. It clearly shows that 20% weightage will be given to HOTS questions. Whereas 55% of questions will be easy to solve.
| 1. | Exhibit memory of previously learned material by recalling facts, terms, basic concepts, and answers. Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas | 44 | 55 |
| 2. | Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way. | 20 | 25 |
| 3. | Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations | 16 | 20 |
| Present and defend opinions by making judgments about information, the validity of ideas, or quality of work based on a set of criteria. | |||
| Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions | |||
| 80 | 100 |
- No. chapter-wise weightage. Care to be taken to cover all the chapters
- Suitable internal variations may be made for generating various templates keeping the overall weightage to different forms of questions and typology of questions the same.
Choice(s): There will be no overall choice in the question paper. However, 33% of internal choices will be given in all the sections
| Periodic Tests ( Best 2 out of 3 tests conducted) | 10 Marks |
| Mathematics Activities | 10 Marks |
12 Maths Prescribed Books
- Mathematics Part I – Textbook for Class XII, NCERT Publication
- Mathematics Part II – Textbook for Class XII, NCERT Publication
- Mathematics Exemplar Problem for Class XII, Published by NCERT
- Mathematics Lab Manual class XII, published by NCERT
Test Generator
Create question paper PDF and online tests with your own name & logo in minutes.
Question Bank, Mock Tests, Exam Papers, NCERT Solutions, Sample Papers, Notes
Related Posts
- CBSE Reduced Syllabus Class 10 (2020-21)
- CBSE Class 10 Maths Sample Paper 2020-21
- Case Study Class 10 Maths Questions
- Competency Based Learning in CBSE Schools
- Class 11 Physical Education Case Study Questions
- Class 11 Sociology Case Study Questions
- Class 12 Applied Mathematics Case Study Questions
- Class 11 Applied Mathematics Case Study Questions
1 thought on “Class 12 Maths Case Study Questions”
plz provide solutions too
Leave a Comment
Save my name, email, and website in this browser for the next time I comment.
Class 12 Maths: Case Study Based Questions PDF Download
- Post author: studyrate
- Post published:
- Post category: 12 board / Class 12
- Post comments: 0 Comments
You must practice some good Case Study questions of Class 12 Maths to boost your preparation to score 95+% on Boards. In this post, you will get Case Study Questions of All Chapters which will come in CBSE Class 12 Maths Board Exams.
Join our Telegram Channel, there you will get various e-books for CBSE 2024 Boards exams for Class 9th, 10th, 11th, and 12th.
We have provided here Case Study questions for the Class 12 Maths exams. You can read these chapter-wise Case Study questions. Prepared by subject experts and experienced teachers. The answer key is also provided so that you can check the correct answer for each question. Practice these questions to score well in your Board Final exams.
We are providing Case Study questions for class 12 Biology based on the latest syllabi. There is a total of 13 chapters included in CBSE class 12 Maths exams. Students can practice these questions for concept clarity and score better marks in their exams.
Table of Contents
CBSE Class 12th – MATHS : Chapterwise Case Study Question & Solution
CBSE will ask two Case Study Questions in the CBSE class 12 maths questions paper. Question numbers 15 and 16 are case-based questions where 5 MCQs will be asked based on a paragraph. Each theme will have five questions and students will have a choice to attempt any four of them.
Case Study Based Questions for Class 12 Maths
Class 12 Physics Case Study Questions Class 12 Chemistry Case Study Questions Class 12 Biology Case Study Questions Class 12 Maths Case Study Questions
Books for Class 12 Maths
Strictly as per the new term-wise syllabus for Board Examinations to be held in the academic session 2022-23 for class 12 Multiple Choice Questions based on new typologies introduced by the board- Stand-Alone MCQs, MCQs based on Assertion-Reason Case-based MCQs. Include Questions from CBSE official Question Bank released in April 2022 Answer key with Explanations What are the updates in the book: Strictly as per the Term wise syllabus for Board Examinations to be held in the academic session 2022-23. Chapter-wise -Topic-wise Multiple choice questions based on the special scheme of assessment for Board Examination for Class 12th.
Class 12 Maths Syllabus 2022-23
Unit-I: Relations and Functions
1. Relations and Functions (15 Periods)
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
2. Inverse Trigonometric Functions (15 Periods)
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Unit-II: Algebra
1. Matrices (25 Periods)
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Oncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants 25 Periods
Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Unit-III: Calculus
1. Continuity and Differentiability (20 Periods)
Continuity and differentiability, chain rule, derivative of inverse trigonometric functions, 𝑙𝑖𝑘𝑒 sin −1 𝑥 , cos −1 𝑥 and tan −1 𝑥, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives (10 Periods)
Applications of derivatives: rate of change of bodies, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as reallife situations).
3. Integrals (20 Periods)
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals (15 Periods)
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
5. Differential Equations (15 Periods)
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
Unit-IV: Vectors and Three-Dimensional Geometry
1. Vectors (15 Periods)
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
2. Three – dimensional Geometry (15 Periods)
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming (20 Periods)
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability 30 (Periods)
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
You Might Also Like
Mcq questions class 12 economics chapter 5 government budget and the economy with answers, class 12 physics assertion reason questions chapter 2 electrostatic potential and capacitance, class 12 physics assertion reason questions chapter 3 current electricity, leave a reply cancel reply.
Save my name, email, and website in this browser for the next time I comment.
This site uses Akismet to reduce spam. Learn how your comment data is processed .
The Topper Combo Flashcards
- Contains the Latest NCERT in just 350 flashcards.
- Colourful and Interactive
- Summarised Important reactions according to the latest PYQs of NEET(UG) and JEE
No thanks, I’m not interested!
RD Sharma solutions for Class 12 Maths chapter 18 - Maxima and Minima [Latest edition]
Online mock tests.
Advertisements
Solutions for chapter 18: maxima and minima.
Below listed, you can find solutions for Chapter 18 of CBSE, Karnataka Board PUC RD Sharma for Class 12 Maths.
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.1 [Page 7]
f(x) = 4x 2 + 4 on R .
f(x) = - (x-1) 2 +2 on R ?
f(x)=| x+2 | on R .
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x)=2x 3 +5 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = 16x 2 \[-\] 16x + 28 on R ?
f(x) = x 3 \[-\] 1 on R .
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.2 [Page 16]
f(x) = (x \[-\] 5) 4 .
f(x) = x 3 \[-\] 3x .
f(x) = x 3 (x \[-\] 1) 2 .
f(x) = (x \[-\] 1) (x+2) 2 .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = x 3 \[-\] 6x 2 + 9x + 15 .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = x 3 (2x \[-\] 1) 3 .
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.3 [Page 31]
f(x) = x 4 \[-\] 62x 2 + 120x + 9.
f(x) = x 3 \[-\] 6x 2 + 9x + 15
f(x) = (x - 1) (x + 2) 2 .
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xe x .
`f(x) = x/2+2/x, x>0 `.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2) 2 .
`f(x)=xsqrt(1-x), x<=1` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx 2 + x has extreme values at x=1 and x=2. Find a and b ?
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum and minimum values of y = tan \[x - 2x\] .
If f(x) = x 3 + ax 2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.4 [Page 37]
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 1) 2 + 3 in [ \[-\] 3,1] ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the maximum value of 2x 3 \[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.5 [Pages 72 - 74]
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm 3 , which has the minimum surface area?
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the lengths of the two pieces so that the combined area of the circle and the square is minimum?
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m 3 . If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a . Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
A closed cylinder has volume 2156 cm 3 . What will be the radius of its base so that its total surface area is minimum ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Show that among all positive numbers x and y with x 2 + y 2 =r 2 , the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the curve y 2 = 4x which is nearest to the point (2,\[-\] 8).
Find the point on the curve x 2 = 8y which is nearest to the point (2, 4) ?
Find the point on the parabolas x 2 = 2y which is closest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x 2 +7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y 2 = 2x which is at a minimum distance from the point (1, 4).
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
The total area of a page is 150 cm 2 . The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.6 [Page 80]
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write sufficient conditions for a point x = c to be a point of local maximum.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the minimum value of f(x) = x x .
Write the maximum value of f(x) = x 1 /x .
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
RD Sharma solutions for Class 12 Maths Chapter 18 Maxima and Minima Exercise 18.7 [Pages 80 - 82]
The maximum value of x 1 /x , x > 0 is __________ .
none of these
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
`ab<c^2/4`
`ab>=c^2/4`
`ab>=c/4`
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
For the function f(x) = \[x + \frac{1}{x}\]
x = 1 is a point of maximum
x = \[-\] 1 is a point of minimum
maximum value > minimum value
maximum value < minimum value
Let f(x) = x 3 +3x 2 \[-\] 9x+2. Then, f(x) has _________________ .
a maximum at x = 1
a minimum at x = 1
neither a maximum nor a minimum at x = - 3
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
\[\frac{1}{2}\]
\[\frac{1}{4}\]
\[\frac{3}{4}\]
Let f(x) = (x \[-\] a) 2 + (x \[-\] b) 2 + (x \[-\] c) 2 . Then, f(x) has a minimum at x = _____________ .
\[\frac{a + b + c}{3}\]
\[\sqrt[3]{abc}\]
\[\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}\]
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
\[\frac{1}{8}\]
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r) 2 assumes minimum value at x = ______________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x 2 + x + 1 is _______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
\[ \frac{1}{4}\]
\[- \frac{1}{3}\]
\[\frac{1}{6}\]
\[\frac{1}{5}\]
The point on the curve y 2 = 4x which is nearest to, the point (2,1) is _______________ .
\[1, 2\sqrt{2}\]
If x+y=8, then the maximum value of xy is ____________ .
The least and greatest values of f(x) = x 3 \[-\] 6x 2 +9x in [0,6], are ___________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
\[\frac{\pi}{3}\]
\[\frac{\pi}{4}\]
\[\frac{\pi}{6}\]
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
\[\frac{1}{3}\]
\[\frac{2}{3}\]
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
\[\frac{4}{3}\]
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
\[2\frac{1}{2}\]
\[3\frac{1}{3}\]
f(x) = 1+2 sin x+3 cos 2 x, `0<=x<=(2pi)/3` is ________________ .
Minimum at x =\[\frac{\pi}{2}\]
Maximum at x = sin \[- 1\] ( \[\frac{1}{\sqrt{3}}\])
Minimum at x = \[\frac{\pi}{6}\]
Maximum at `sin^-1(1/6)`
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
\[- \frac{1}{4}\]
Let f(x) = 2x 3 \[-\] 3x 2 \[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The minimum value of x log e x is equal to ____________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
RD Sharma solutions for Class 12 Maths chapter 18 - Maxima and Minima
Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. RD Sharma solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 18 (Maxima and Minima) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.
Concepts covered in Class 12 Maths chapter 18 Maxima and Minima are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies or Quantities, Introduction to Applications of Derivatives.
Using RD Sharma Class 12 Maths solutions Maxima and Minima exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in RD Sharma Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Class 12 Maths students prefer RD Sharma Textbook Solutions to score more in exams.
Get the free view of Chapter 18, Maxima and Minima Class 12 Maths additional questions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.
- Maharashtra Board Question Bank with Solutions (Official)
- Balbharati Solutions (Maharashtra)
- Samacheer Kalvi Solutions (Tamil Nadu)
- NCERT Solutions
- RD Sharma Solutions
- RD Sharma Class 10 Solutions
- RD Sharma Class 9 Solutions
- Lakhmir Singh Solutions
- TS Grewal Solutions
- ICSE Class 10 Solutions
- Selina ICSE Concise Solutions
- Frank ICSE Solutions
- ML Aggarwal Solutions
- NCERT Solutions for Class 12 Maths
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Chemistry
- NCERT Solutions for Class 12 Biology
- NCERT Solutions for Class 11 Maths
- NCERT Solutions for Class 11 Physics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Biology
- NCERT Solutions for Class 10 Maths
- NCERT Solutions for Class 10 Science
- NCERT Solutions for Class 9 Maths
- NCERT Solutions for Class 9 Science
- CBSE Study Material
- Maharashtra State Board Study Material
- Tamil Nadu State Board Study Material
- CISCE ICSE / ISC Study Material
- Mumbai University Engineering Study Material
- CBSE Previous Year Question Paper With Solution for Class 12 Arts
- CBSE Previous Year Question Paper With Solution for Class 12 Commerce
- CBSE Previous Year Question Paper With Solution for Class 12 Science
- CBSE Previous Year Question Paper With Solution for Class 10
- Maharashtra State Board Previous Year Question Paper With Solution for Class 12 Arts
- Maharashtra State Board Previous Year Question Paper With Solution for Class 12 Commerce
- Maharashtra State Board Previous Year Question Paper With Solution for Class 12 Science
- Maharashtra State Board Previous Year Question Paper With Solution for Class 10
- CISCE ICSE / ISC Board Previous Year Question Paper With Solution for Class 12 Arts
- CISCE ICSE / ISC Board Previous Year Question Paper With Solution for Class 12 Commerce
- CISCE ICSE / ISC Board Previous Year Question Paper With Solution for Class 12 Science
- CISCE ICSE / ISC Board Previous Year Question Paper With Solution for Class 10
- Entrance Exams
- Video Tutorials
- Question Papers
- Question Bank Solutions
- Question Search (beta)
- More Quick Links
- Privacy Policy
- Terms and Conditions
- Shaalaa App
- Ad-free Subscriptions
Select a course
- Class 1 - 4
- Class 5 - 8
- Class 9 - 10
- Class 11 - 12
- Search by Text or Image
- Textbook Solutions
- Study Material
- Remove All Ads
- Change mode
Book a Trial With Our Experts
Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
Thank you for registering.
One of our academic counsellors will contact you within 1 working day.
Click to Chat
- 1800-5470-145
- +91 7353221155
- Login | Register
- My Classroom
- My Self Study Packages
- Batch Discussion
- My Forum Activity
- Refer a Friend
- Edit Profile
- Add Question
- Add Paragraph
- Search Coupon
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Register Now and Win Upto 25% Scholorship for a Full Academic Year !
Enter your details.
Registration done!
Sit and relax as our customer representative will contact you within 1 business day
Mobile Verification
OTP to be sent to Change
- Junior Hacker
- Junior Hacker New
- Self Study Packages
- JEE Advanced Coaching
- 1 Year Study Plan
- Rank Predictor
- Paper Pattern
- Important Books
- Sample Papers
- Past Papers
- Preparation Tips
- Latest News
- JEE Main Exams
- Online Coaching
- Branch Predictor
- JEE Main Syllabus
- Past Year Papers
- Math Preparation Tips
- IIT JEE Exam Details
- JEE Syllabus
- IIT JEE Toppers Tips
- IIT JEE Preparation Tips
- IIT JEE Preparation Tips for Class 11
- IIT JEE Preparation Tips for Class 9
- IIT JEE Preparation Tips for Class 8
- IIT JEE Preparation Time Table
- IIT JEE Online Coaching
- Correspondence Course For IIT JEE
- IIT JEE Coaching after 10th
- IIT JEE Coaching For Foundation Classes
- JEE Coaching Institutes
- IIT JEE Coaching in Kota
- IIT JEE Coaching Institutes In Kota
- BITSAT Examination
- View complete IIT JEE Section
- View All Engineering Exams
- Top Engineering Colleges
- Top Engineering Branches
- Engineering Exam Calendar
- NEET Entrance Exam
- NEET Online Coaching
- NEET Preparation Tips
- Participating States
- AIIMS Examination
- AIIMS Online Coaching
- View all Medical Exams
- Top Medical Colleges
- Medical Exam Coaching
- Best Medical Coaching In Kota
- Medical Exam Calendar
- NTSE Examination
- Notifications
- Application
- Important Dates
- Eligibility
- Study Material
- KVPY Examination
- Olympiads Examination
- Indian National Mathematics Olympiad
- Physics Olympiad
- Chemistry Olympiad
- Biology Olympiad
- Olympiads Sample Papers
- INMO Papers
- CBSE School Exams
- Solutions for Board Exam
- JEE Advanced
- Karnataka CET
- Manipal UGET
- NCERT Class 12 Solutions
- NCERT Class 11 Solutions
- NCERT Class 10 Solutions
- NCERT Class 9 Solutions
- NCERT Class 8 Solutions
- NCERT Class 7 Solutions
- NCERT Class 6 Solutions
- List of JEE Main & JEE Advanced Books
- R.D. Sharma Solutions PDFâ
- Concepts of Physics by HC Verma for JEE
- HC Verma Solutions Part 1
- HC Verma Solutions Part 2
- Most Scoring Topics in IIT JEE
- IIT JEE Entrance Exam
- Discuss with Colleagues and IITians
- Engineering Entrance Exams
- Branch Ranking of IIT
- Discuss with Askiitians Tutors
- NEET (AIPMT)
- Marks and Rank in IIT JEE
- Top Engineering Colleges in India
- AIEEE Entrance Exam
- Electric Current
- Wave Motion
- Modern Physics
- Thermal Physics
- Electromagnetic Induction
- General Physics
- Electrostatics
- Wave Optics
- Physical Chemistry
- Organic Chemistry
- Inorganic Chemistry
- Trigonometry
- Analytical Geometry
- Differential Calculus
- Integral Calculus
- Magical Mathematics
- Online Tutoring
- View complete NRI Section
- View Complete Study Material
- View Complete Revision Notes
- Ahmadi (FAIPS)
- Khaitan (Carmel School)
IIT JEE Courses
One Year IIT Programme
- Super Premium LIVE Classes
- Top IITian Faculties
- 955+ hrs of Prep
- Test Series & Analysis
Two Year IIT Programme
- 1,835+ hrs of Prep
Crash Course
- LIVE + Pre Recorded Sessions
- 300+ hrs of Prep
NEET Courses
One Year NEET Programme
- Top IITian & Medical Faculties
- 900+ hrs of Prep
Two Year NEET Programme
- 1,820+ hrs of Prep
- LIVE 1-1 Classes
- Personalized Sessions
- Design your own Courses
- Personalized Study Materials
School Board
Live online classes, class 11 & 12.
- Class 11 Engineering
- Class 11 Medical
Class 9 & 10
Class 6, 7 & 8, test series, jee test series.
- 2 Year Jee Test Series
- 1 Year Jee Test Series
NEET test series
- 2 Year NEET Test Series
- 1 Year NEET Test Series
C.B.S.E test series
- 11 Engineering
- 12 Engineering
Complete Self Study Packages
Full course.
- 2 year NEET
- Chemistry 11th & 12th
- Maths 11th & 12th
- Physics 11th & 12th
- Biology 11th & 12th
- View Complete List
For class 12th
- Chemistry class 12th
- Maths class 12th
- Physics class 12th
- Biology class 12 th
For class 11th
- Chemistry class 11th
- Maths class 11th
- Physics class 11th
- Biology class 11th
Chapter 18: Maxima and Minima
Maxima and minima.
One of the major applications of derivatives is determination of the maximum and minimum values of a function. We know that the derivative provides the information about the gradient of the graph of a function which can be used to identify the points where the gradient is zero. These points are usually related to the largest and the smallest values of the function. For more details use the links given below:
Click here → Study Material of Maxima and Minima
Click here → Revision Notes of Maxima and Minima
TOP Your EXAMS!
Upto 50% scholarship on live classes, course features.
- Video Lectures
- Revision Notes
- Previous Year Papers
- Study Planner
- NCERT Solutions
- Discussion Forum
- Test paper with Video Solution
Book Free demo of askIITians Live class
View courses by askiitians.
Design classes One-on-One in your own way with Top IITians/Medical Professionals
Complete Self Study Package designed by Industry Leading Experts
Live 1-1 coding classes to unleash the Creator in your Child
a Complete All-in-One Study package Fully Loaded inside a Tablet!
Ask question.
Get your questions answered by the expert for free
Your Question has been posted!
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
POST QUESTION
Select the tag for question.
Chapter 18: Maxima and Minima – Exercise...
Join NOW to get access to exclusive study material for best results
- Class 12-science
- CBSE Class 12-science Textbook Solutions
- RD SHARMA Solutions
- Chapter 18 Maxima and Minima
Please Login or Sign up to continue!
Class 12-science RD SHARMA Solutions Maths Chapter 18 - Maxima and Minima
Maxima and minima exercise mcq, solution 30.
f'(x) = 2x - 8
Take f'(x) = 0, we get
Again differentiating, we get
f"(x) = 2 > 0 for all real x
Therefore, x = 4 is the point of minima.
The minimum value of f(x) is
f(4) = 16 - 32 + 17 = 1
Solution 32
The maximum value of f(x) is
Solution 33
Taking log on both the sides, we get
From equation (ii), we get
Solution 34
Solution 35
Again differentiating w.r.t x, we get
So, the slope is maximum at x = 1
Solution 36
Now, let's find f(x) at x = 2 or -1
Therefore, x = -1 is the point of local maxima and the maximum value is 11.
Whereas, x = 2 is the point of local minima and the minimum value is -16.
Hence, f(x) has one maximum and one minimum.
Solution 10
Solution 11
Solution 12
Solution 13
Solution 14
Solution 15
Solution 16
Solution 17
Solution 18
Solution 19
Solution 20
Solution 21
Solution 22
Solution 23
Solution 24
Solution 25
Solution 26
Solution 27
Solution 28
Solution 29
The idea behind introducing this question is to identify whether the student has understood the logical sequences that should be performed while performing any type of calculation. These are of very basic level questions where the focus is to understand the importance of mathematical operator, based on its position. If a , — x, v sign comes before a number then it has entirety different significance fromwhat it gives when comes after the number. This is where the practice of BODIVIAS is required.
Maxima and Minima Exercise Ex. 18.1
Maxima and Minima Exercise Ex. 18.2
Maxima and Minima Exercise Ex. 18.3
Solution 1(i).
Solution 1(ii)
Solution 1(iii)
Solution 1(iv)
Solution 1(v)
Solution 1(vi)
Solution 1(vii)
Solution 1(viii)
Solution 1(ix)
Solution 1(x)
Solution 1(xi)
Solution 1(xii)
Solution 2(i)
Solution 2(ii)
Solution 2(iii)
Take f'(x) = 0
Differentiating f'(x) w.r.t x, we get
Maxima and Minima Exercise Ex. 18.4
Maxima and Minima Exercise Ex. 18.5
Solution 37
Solution 38.
Solution 39
Solution 40
Solution 41
Solution 42
Solution 43
Solution 44
Solution 45
Let r and h be the radius and height of the cylinder.
From (i), we get
Taking O as the centre of the circle, join OE, OF and OD such that
OE = OF = OD = r (radius)
Similarly, AF = r cot x
P = AB + BC + CA
= AE + EC + BD + DC + AF + BF
Taking second derivative of P, we get
Maxima and Minima Exercise Ex. 18VSAQ
CBSE Class 10 Maths MIQs, Subjective Questions & More
Once you complete your vast syllabus, you can continue to revise the concepts learned in your class. Once you complete your vast syllabus, you can continue to revise the concepts learned in your class. Once you complete your vast syllabus, you can continue to revise the concepts learned in your class. Once you complete your vast syllabus, you can continue to revise the concepts learned in your class.
Once you complete your vast syllabus, you can continue to revise the concepts learned in your class.
Frequently Asked Questions
Get in Touch
Call / Whats App
- - Class 11 Science
- - Class 11 Commerce
- - Class 12 Science
- - Class 12 Commerce
- Maxima and Minima
As the name suggests, this topic is devoted to the method of finding the maximum and the minimum values of a function in a given domain. It finds application in almost every field of work, and in every subject. Let’s find out more about the maxima and minima in this topic.
Suggested Videos
Some day-to-day applications are described below:
- To an engineer – The maximum and the minimum values of a function can be used to determine its boundaries in real-life. For example, if you can find a suitable function for the speed of a train; then determining the maximum possible speed of the train can help you choose the materials that would be strong enough to withstand the pressure due to such high speeds, and can be used to manufacture the brakes and the rails etc. for the train to run smoothly.
- To an economist – The maximum and the minimum values of the total profit function can be used to get an idea of the limits the company must put on the salaries of the employees, so as to not go in loss.
- To a doctor – The maximum and the minimum values of the function describing the total thyroid level in the bloodstream can be used to determine the dosage the doctor needs to prescribe to different patients to bring their thyroid levels to normal.
Types of Maxima and Minima
The maxima or minima can also be called an extremum i.e. an extreme value of the function. Let us have a function y = f(x) defined on a known domain of x. Based on the interval of x, on which the function attains an extremum, the extremum can be termed as a ‘local’ or a ‘global’ extremum. Let’s understand it better in the case of maxima.
Browse more Topics under Application Of Derivatives
- Rate of Change of Quantities
- Approximations
- Increasing and Decreasing Functions
- Tangents and Normals
- Local Maxima
A point is known as a Local Maxima of a function when there may be some other point in the domain of the function for which the value of the function is more than the value of the local maxima, but such a point doesn’t exist in the vicinity or neighborhood of the local maxima. You can also understand it as a maximum value with respect to the points nearby it.
Global Maxima
A point is known as a Global Maxima of a function when there is no other point in the domain of the function for which the value of the function is more than the value of the global maxima. Types of Global Maxima:
- Global maxima may satisfy all the conditions of local maxima. You can also understand it as the Local Maxima with the maximum value in this case.
- Alternately, the global maxima for an increasing function could be the endpoint in its domain ; as it would obviously have the maximum value. In this case, it isn’t a local maximum for the function.
Similarly, the local and the global minima can be defined. Look at the graph below to identify the different types of maxima and minima.
Stationary Points
A stationary point on a curve is defined as one at which the derivative vanishes i.e. a point (x 0 , f(x 0 )) is a stationary point of f(x) if \({[\frac{df}{dx}]_{x = x_0} = 0}\). Types of stationary points:
- Local Minimas
- Inflection Points
We won’t discuss inflection points here. As of now though, you must note that all the points of extremum are stationary points.
Proof: I’ll prove the above statement for the case of a Local Maxima. Others will simply follow from this. Let us have a function y = f(x) that attains a Local Maximum at point x = x 0 . Near the extremum point, the curve will look something like this:
Clearly, the derivative of the function has to go to 0 at the point of Local Maximum; otherwise, it would never attain a maximum value with respect to its neighbors.
The Second Derivative Test
This test is used to determine whether a stationary point is a Local Maxima or a Local Minima. Whether it is a global maxima/global minima can be determined by comparing its value with other local maxima/minima. Let us have a function y = f(x) with x = x 0 as a stationary point. Then the test says:
- If \({[\frac{d^2f}{d^2x}]_{x = x_0} < 0}\), then x = x 0 is a point of Local Maxima.
- If \({[\frac{d^2f}{d^2x}]_{x = x_0} > 0}\), then x = x 0 is a point of Local Minima.
- If for x > x 0 , \({[\frac{df}{dx}]_{x = x_0} < 0 }\) and for x < x 0 , \({[\frac{df}{dx}]_{x = x_0} > 0}\) i.e. the function is increasing for x < x 0 and decreasing for x > x 0 ; we can conclude that x = x 0 is a point of Local Maxima.
- Similarly, if for x > x 0 , \({[\frac{df}{dx}]_{x = x_0} > 0 }\) and for x < x 0 , \({[\frac{df}{dx}]_{x = x_0} < 0}\) i.e. the function is decreasing for x < x 0 and increasing for x > x 0 ; we can conclude that x = x 0 is a point of Local Minima.
The proof of the third case can be understood by looking at Fig 1. above for local maxima. Similarly, for local minima, one can get:
Proof of the Second Derivative Test
We’ll prove the test for the case of a Local Minima. The proof for a Local Maxima will follow in a similar fashion. Take a look at the Fig 2. above. One can see that the slope of the tangent drawn at any point on the curve i.e. \({\frac{dt}{dx}}\) changes from a negative value to 0 to a positive value, near the point of local minima. T
his means that the function that is represented by(say) \({f(x) = \frac{dt}{dx}}\) behaves like an increasing function. The condition for a function to be increasing is: $$ {\frac{df}{dx} > 0}{\text{ i.e.} \frac{d^2y}{d^2x} > 0} $$ This confirms that the function will have a local minima if the first derivative is 0, and the second derivative is positive at that point.
Solved Examples for You on Maxima and Minima
Question 1 : Find the local maxima and minima for the function y = x 3 – 3x + 2.
Answer : We’ll need to find the stationary points for this function, for which we need to calculate \({\frac{df}{dx}}\). We’ll proceed as follows:
$$ { y = x^3 – 3x +2} $$ $${\frac{dy}{dx} = 3x^2 – 3} $$
At stationary points, \({\frac{dy}{dx} = 0}\). Thus, we have;
$$ {3x^2 – 3 = 0} $$ $$ {3(x^2 – 1) = 0} $$ $$ {(x – 1)(x + 1) = 0} $$ $$ { \text{ x = 1 / x = -1}} $$
Now we have to determine whether any of these stationary points are extremum points. We’ll use the second derivative test for this:
$${\frac{dy}{dx} = 3x^2 – 3} $$ $${\frac{d^2y}{d^2x} = 6x } $$
- For x = 1; \( {\frac{d^2y}{d^2x} = 6/times{1} = 6}\), which is positive. Thus the point (1, y(x = 1)) is a point of Local Minima.
- For x = -1; \( {\frac{d^2y}{d^2x} = 6/times{-1} = -6}\), which is positive. Thus the point (-1, y(x = -1)) is a point of Local Maxima.
We can see from the graph below to verify our calculations:
This concludes our discussion on this topic of maxima and minima.
Question 2: What are relative maxima and relative minima?
Answer: Finding out the relative maxima and minima for a function can be done by observing the graph of that function. A relative maxima is the greater point than the points directly beside it at both sides. Whereas, a relative minimum is any point which is lesser than the points directly beside it at both sides.
Question 3: How to find out the absolute maxima of a function?
Answer: Finding the absolute maxima:
Firstly, find out all the critical numbers of the function within the interval [a, b].
Then, plug in every single critical number from the first step into the function i.e. f(x).
Plugin the ending points that are (a) and (b) into the function f(x).
Finally, the biggest value is the absolute maxima and the lowest value is the absolute minima.
Question 4: What is the absolute maxima?
Answer: The biggest value that a mathematical function can consume over its whole curve. The absolute maxima on the graph takes place at x = d, and the absolute minima of that graph takes place at x = a.
Question 5: What are the local and global maxima and minima?
Customize your course in 30 seconds
Which class are you in.
Application of Derivatives
Leave a reply cancel reply.
Your email address will not be published. Required fields are marked *
Download the App
Talk to our experts
1800-120-456-456
- RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.3) Exercise 18.3
- RD Sharma Solutions
RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.3) Exercise 18.3 - Free PDF
Free PDF download of RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima Exercise 18.3 solved by Expert Mathematics Teachers on Vedantu. All Chapter 18 - Maxima and Minima Ex 18.3 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.
Topics included in Class 12 RD Sharma Solutions for Chapter 18, that is, Maxima and Minima
Class 12 Chapter 18, that is, Maxima and Minima of RD Sharma include topics like maximum and minimum values of a function, definition and meaning of maximum and minimum, Local Maxima and Minima, first derivative test for Local Maxima and Minima, higher-order derivative test, theorem and algorithm based on the higher derivative test, point of inflexion and inflexion, properties of Maxima and Minima, maximum and minimum values in a closed interval and applied problems on Maxima and Minima.
What are the features of RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima?
The main features of RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima are that RD Sharma Class 12 contains detailed theory and illustrations, an algorithmic approach, a large number of graded illustrative examples and exercises and a summary of concepts and formulae. These features make RD Sharma one of the best books to study for Class 12 students. They act as revision notes for the students because students get concept clarity, example problems and formulae all in one single place.
FAQs on RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.3) Exercise 18.3
1. What topics are covered in Exercise 18.3 of RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima?
Chapter 18, that is, Maxima and Minima of RD Sharma Class 12 is very interesting and easy to understand the chapter. The exercise is concerned with some major parts of the chapter. The major topics covered in Chapter 18, that is, Maxima and Minima are higher-order derivative tests, theorems, and algorithms based on the higher derivative tests, points of inflexion, points of inflexion, and properties of Maxima and Minima. The questions from these topics are asked in exams as well, therefore, the students must prepare this chapter well.
2. From where can I download the offline version of RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima?
RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima, exercise 18.3 can be downloaded as a PDF file from the official website of Vedantu. The students will register themselves at the official website of Vedantu through their email address and they can then download the PDF file of the solutions available on the website. This PDF file will be available with the students and they can assess these files in an offline mode as well.
3. How can RD Sharma Class 12 Solutions for Chapter 18 help me in my exams?
RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima are very important for the students. The RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima is a book written in a simple language and it provides students with an easy way to prepare for the exams. The RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima involve chapter-wise solutions to each question. The questions available in RD Sharma are also asked in final exams, therefore, students can score better in exams by practising these questions.
4. Why should I prefer Vedantu for downloading RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima?
Students should download RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima from Vedantu because of some added benefits that the students may get:
The PDF file of RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima are free of cost at the official website of Vedantu.
The RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima contain important questions which may be asked in the examination.
The RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima covers all the important concepts and therefore, students will be able to revise the full course in a very short period.
5. How many exercises are present in RD Sharma Class 12 Solutions for Chapter 18 Maxima and Minima?
The RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima contains a total of 5 exercises. The first exercise of Chapter 18 covers questions wherein we need to find the maximum and minimum values. The second exercise of Chapter 18 covers the questions to calculate the points of Local Maxima and Minima of the given functions. The third exercise prepares students to calculate Local Maxima and Minima, corresponding Local Maxima and Minima, and Local extremism. The fourth exercise allows students to find the absolute maximum and minimum values of the given function. The last exercise of the chapter allows students to apply the concept of integration.
- [email protected]
- Search for: Search
Tapati's Classes
Maxima and Minima. Previous Years Board Questions (2000 to Sem 1-2021) with Solutions of ISC Class 12 Mathematics.
Solutions of Maxima and Minima – Previous Years Board Questions – (2000 to 2020) – ISC – Class 12 – Mathematics
Maxima and Minima. Click here to open solutions
Scroll Down for solutions of Sem1-2021.
| apati’s Classes – Online LIVE Maths Classes
Contact: [email protected] |
|
IMAGES
VIDEO
COMMENTS
Class 12 Maths question paper will have 1-2 Case Study Questions. These questions will carry 5 MCQs and students will attempt any four of them. As all of these are only MCQs, it is easy to score good marks with a little practice. Class 12 Maths Case Study Questions are available on the myCBSEguide App and Student Dashboard.
CBSE Class 12 Maths Case Study Questions With Solutions CBSE Class 12 Mathematics Case Study Questions are introduced this year in the updated CBSE Board Exam Pattern. According to that this year the candidates need to prepare for the case study problems along with the questions that are legacy of CBSE Board.
We have provided here Case Study questions for the Class 12 Maths term 2 exams. You can read these chapter-wise Case Study questions for your term 2 science paper. These questions are prepared by subject experts and experienced teachers. Answer key is also provided so that you can check the correct answer for each question. Practice these questions to score well in your term 2 exams.
Students can get free Updated 2023-24 RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima here. Visit BYJU'S to download the PDF version of RD Sharma Solutions Class 12.
Get free RD Sharma Solutions for Class 12 Maths Chapter 18 Maxima and Minima solved by experts. Available here are Chapter 18 - Maxima and Minima Exercises Questions with Solutions and detail explanation for your practice before the examination
In RD Sharma Solutions for Class 12th Maths Chapter 18 maxima and minima, students first obtain knowledge on how to describe maximum and minimum values. Maximum is a point at which the value of a function is greatest, whereas minimum in maths is a point at which the value of a function is less than or equal to the value at any nearby point (local point) or an absolute minimum. Along with this ...
Free PDF download of RD Sharma Solutions for Class 12 Maths Chapter 18 - Maxima and Minima solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 18 - Maxima and Minima Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced), NEET, Engineering and Medical entrance exams.
Free PDF download of RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima Exercise 18.2 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 18 - Maxima and Minima Ex 18.2 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise complete Syllabus and Score More marks.
Maxima and Minima One of the major applications of derivatives is determination of the maximum and minimum values of a function. We know that the derivative provides the information about the gradient of the graph of a function which can be used to identify the points where the gradient is zero. These points are usually related to the largest and the smallest values of the function. For more ...
RD Sharma Solutions for Class 12-science Maths CBSE Chapter 18: Get free access to Maxima and Minima Class 12-science Solutions which includes all the exercises with solved solutions. Visit TopperLearning now!
Download free pdf of RD Sharma Class 12 Maths chapter Maxima and Minima CSBQ solutions to see shortcut methods and simple formulas to get more marks in your exams.
Get Free RD Sharma Class 12 Solutions Chapter 18 Ex 18.1. Maxima and Minima Class 12 Maths RD Sharma Solutions are extremely helpful while doing your homwork or while preparing for the exam.
Maxima and Minima As the name suggests, this topic is devoted to the method of finding the maximum and the minimum values of a function in a given domain. It finds application in almost every field of work, and in every subject. Let's find out more about the maxima and minima in this topic.
Free PDF download of RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima Exercise 18.1 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 18 - Maxima and Minima Ex 18.1 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise complete Syllabus and Score More marks.
Download free pdf of RD Sharma 12th maths chapter 17 Maxima and Minima exercise 17.5 solutions to see shortcut methods and simple formulas to get more marks in your exams.
The RD Sharma Class 12 Solutions for Chapter 18, that is, Maxima and Minima contains a total of 5 exercises. The first exercise of Chapter 18 covers questions wherein we need to find the maximum and minimum values. The second exercise of Chapter 18 covers the questions to calculate the points of Local Maxima and Minima of the given functions.
Solutions of Maxima and Minima - Previous Years Board Questions - (2000 to 2020) - ISC - Class 12 - Mathematics Maxima and Minima. Click here to open solutions Scroll Down for solutions of Sem1-2021.
Updated 2023-24 RD Sharma Solutions for Class 12 Maths Chapter 18 - Maxima and Minima Exercise 18.5 covers the topic of applied problems on maxima and minima of a given function, with a descriptive solution.
Download free pdf of RD Sharma 12th maths chapter 17 Maxima and Minima MCQ solutions to see shortcut methods and simple formulas to get more marks in your exams.
Apart from catering students preparing for JEE Mains and NEET, PW also provides study material for each state board like Uttar Pradesh, Bihar, and others. Question of Class 12-Maxima and Minima : From the figure for the function y = f (x) We find that the function gets local maximum and local minimum at the points.
The RD Sharma solutions Class 12 Maxima and Minima helps all the students in doing the exam preparation for final exams. Our highly qualified subject matter experts who have years of experience in the education industry have created the Maxima and Minima solutions in a stepwise manner to ensure that the students go through them and grasp all the important concepts. Not only for studying but ...
6 Marks Questions. Filed Under: CBSE Tagged With: Class 12 Maths, Maths Maxima and Minima. Free Resources. RD Sharma Class 12 Solutions. RD Sharma Class 11. RD Sharma Class 10. RD Sharma Class 9.
Class 12 RD Sharma Solutions - Chapter 18 Maxima and Minima - Exercise 18.5 | Set 1. Last Updated : 08 Dec, 2022. Question 1. Determine two positive numbers whose sum is 15 and the sum of whose squares is minimum. Solution: Let us assume the two positive numbers are x and y, And it is given that x + y = 15 …..