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RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.2) Exercise 18.2

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MVSAT Dec 2023

RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.2) Exercise 18.2 - Free PDF

Free PDF of RD Sharma Class 12 Solutions Chapter 18 – Maxima and Minima Exercise 18.2 solved by expert Mathematics teachers is available for download on Vedantu.com. All Chapter 18 – Maxima and Minima Ex 18.2 Questions with Solutions for RD Sharma Class 12 Maths will 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 on Vedantu.com.

 

Maxima and minima are some of the important concepts taught in Class 12. It is Chapter 18 in RD Sharma’s book while in the NCERT textbook it is discussed in depth under Chapter 6 called application of derivatives.

 

As this concept is extremely vast, RD Sharma has given it utmost importance by separating it from the rest of the concepts. Maxima and minima are the largest and smallest values of the function, either within a given range or on the entire domain. It teaches in-depth about the method to calculate the maximum and the minimum value of a function that is the highest and the lowest points in the graph.

Competitive Exams after 12th Science

RD Sharma Class 12 Solutions Chapter 18 – Maxima and Minima

Important Definitions Covered in This Chapter

1. Let f be a function defined on an interval I. Then,


(a) f is said to have a maximum value in I, if there exists a point c in I such that f(c)>f(x), for all x ∈ I. The number f(c) is called the maximum value of f in I and the point c is called a point of maximum value of f in I.


(b) f is said to have a minimum value in I, if there exists a point c in I such that f(c) < f(x), for all x ∈ I. The number f(c), in this case, is called the minimum value of f in I and the point c, in this case, is called a point of minimum value of f in I.


(c) f is said to have an extreme value in I if there exists a point c in I such that f(c) is either a maximum value or a minimum value of f in I. The number f(c), in this case, is called an extreme value of f in I and the point c is called an extreme point.


2.  Let f be a real-valued function and let c be an interior point in the domain of f. Then,


(a) c is called a point of local maxima if there is an h > 0 such that f(c) ≥ f(x), for all x in (c – h, c + h) x ≠ c, the value f(c) is called the local maximum value of f.


(b) c is called a point of local minima if there is an h > 0 such that f(c) ≤ f(x), for all x in (c – h, c + h), the value f(c) is called the local minimum value of f .


Important Theorems 

Let f be a function defined on an open interval I. Suppose c ∈ I be any point. If f has a local maxima or local minima at x = c, then either f ′(c) = 0 or f is not differentiable at c.


(First Derivative Test) Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then,


(i) If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.


(ii) If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.


(iii) If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called point of inflection.

(Second Derivative Test) Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then,


(i) x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0. The value f(c) is the local maximum value of f.


(ii) x = c is a point of local minima if f′(c)=0 and f″(c)>0. In this case, f(c) is the local minimum value of f.


(iii) The test fails if f ′(c) = 0 and f ″(c) = 0.


Theorem: Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.


Theorem:  Let f be a differentiable function on a closed interval I and let c be any interior point of I. Then,


(i) f ′(c) = 0 if f attains its absolute maximum value at c.


(ii) f ′(c) = 0 if f attains its absolute minimum value at c.

 


FAQs on RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.2) Exercise 18.2

1. Define maxima and minima.

Maxima and minima are the largest and smallest values of the function, either within a given range or on the entire domain. It is the maximum and the minimum value of a function that is the highest and the lowest points in the graph. This is further cleared in RD Sharma practice questions which explain how to use the various theorems and definitions discussed in this chapter.

2. What are the important topics discussed in maxima and minima?

The important topics discussed are as follows:


First and second derivative tests, maximum and minimum values of a function in a closed interval, and important theorems are explained in detail, giving a comprehensive understanding of the vast concepts and minute details needed to ace this part of geometry.

3. Why study maxima and minima?

Maxima and minima are extremely important to study linear algebra and game theory. It helps in calculating variations which help to find the extreme values of a function. These values can be calculated by using 1st or 2nd derivative methods. This part of algebra holds a significant weightage in the examination and when the concept is understood in depth, it can fetch you a good score easily.

4. Where can I find RD Sharma study material of maxima and minima?

The study material and exercise solutions of the addition of books are easily available on Vedanta‘s website and the study material is available for free download so that students can take that PDF with them anywhere and can have a distraction-free study session at any time.

5. Are the questions given in RD Sharma difficult to solve?

The practice questions given in the RD Sharma textbook are made to increase the knowledge of the students about the important topics taught in class. RD Sharma is based on the CBSE curriculum, giving the students a much-needed boost to prepare exceedingly well. Yes, the questions are tough, but if you follow the sequence, you will be able to solve it as RD Sharma increases the toughness in a progressing manner.