# RD Sharma Class 12 Chapter 18 Solutions (Maxima And Minima)

RD Sharma Class 12 Maths Solutions Chapter 18 Maxima and Minima will prepare you to answer questions from elementary level to complex ones. These RD Sharma Solutions cover all the topics that chapter entails. Various topics discussed in Chapter 18 Maxima and Minima are Maximum, Minimum, Local Maxima, Local Minima, theorem, first derivative test for local maxima and local minima and related theorems, higher-order derivative test, point of injection, properties of Maxima and Minima, point of inflexion, maximum and minimum values in a closed interval, and applied problems on maxima and minima.

Chapter Maxima and Minima comprise 137 questions divided into seven exercises. These questions test your knowledge of the chapter and its concepts that you can apply in solving most basic to highly advanced problems. Upon solving these questions, you’ll find how important Maxima and Minima are in solving complex mathematical problems.

Instasolv provides all the solutions comprising seven exercises, arranged in exactly the same order as you’d find in RD Sharma Class 12 Solutions Maths Chapter 18 Maxima and Minima. These problems are a level-up of those you find in textbooks, preparing you at every stage to appear for CBSE and other competitive examinations.

## Topics Discussed in RD Sharma Class 12 Maths Solutions for Chapter 18 Maxima and Minima

**Maximum – **Let the real function f(x) is defined on an I interval. Then, f(x) will have the maximum value in the interval I if and only if there is a point in the interval I such that f(x) ≤ f(a) for x ϵ I.

In such a case, f(c) will be known as the maximum value of the function f(x) in an I interval. The point c here will be known as the maximum value point of the f function in Interval I.

**Minimum – **f(x) is a real function defined on the interval I. Then f(x) will have the minimum value in I, if there is a point c ϵ I such that f(x) ≥ f(c) for all values of x ϵ I.

In this case, the number f(c) will be known as the minimum value of f(x) in the given interval I. Point c will be known as f’s point of the minimum value in the given interval I.

Unlike maximum or greatest and minimum or least values of a function in a given domain, there are points in the same domain of a function f(x) where the f(x) does not get the minimum or maximum value. However, the values at each of these points will be larger than or less than the values of a function at a neighbouring point.

**Local Maxima**

F(x) is the function that attains a local maximum at the point x = a, if there is a neighbourhood i.e. (a-δ, a+δ) of the value a such that

f(x) < f(a) for all the values of x ϵ (a-δ, a+δ), x ≠ a.

OR, f(x) – f(a) < 0 for all the values of x ϵ (a – δ, a + δ), x ≠ a.

In such cases, the local maximum of f(x) at x = a will be f(a).

**Local Minima**

f(x) function that attains a local minimum at the point x = a, if there is a neighbourhood i.e. (a-δ, a+δ) of the value a such that

f(x) > f(a) for all the values of x ϵ (a-δ, a+δ), x≠a.

OR, f(x) – f(a) > 0 for all the values of x ϵ (a-δ, a+δ), x≠a.

### Discussion of exercises in RD Sharma Class 12 Maths Solutions for Chapter 18 Maxima and Minima

- The first exercise 18.1 discusses how you can find the maximum and minimum values without leveraging derivatives.
- The second exercise 18.2 discusses the methodology to use while calculating the points of local maxima and local minima of the given functions using only the first-order derivative test.
- The third exercise 18.3 prepares you to find the local maxima, local minima, corresponding local maxima, corresponding local minima, local extremism, etc for a given function f(x).
- The fourth exercise 18.4 teaches you to find the absolute maximum and minimum values of the given function.
- The fifth exercise 18.5 discusses the methodology you use in applying the concept of the entire chapter in integration with all the previous chapters of Class 12 mathematics.
- The rest of the exercises covers complex topics of the Chapter 18 Maxima And Minima

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