RD Sharma Class 12 Chapter 15 Solutions (Mean Value Theorems)
RD Sharma Class 12 Maths Solutions Chapter 15 discusses problems in which you’ll learn the basics of Mean Value Theorems. These RD Sharma Solutions are curated in easy language for you to understand the concepts and the methodology to apply theorems to solve mathematical problems in CBSE and competitive examinations. Various topics discussed in the chapter are Rolle’s Theorem, Geometrical Interpretation of Rolle’s Theorem, Algebraic Interpretation of Rolle’s Theorem, Lagrange’s Mean Value Theorem, and Geometrical Representation of Lagrange’s Mean Value Theorem.
The solutions are listed in total 4 exercises comprising 96 questions. The chapter teaches you about how to check the applicability of Rolle’s theorem or verifying Rolle’s Theorem for a given function on a defined interval, or proving the validity of Lagrange’s theorem by solving mathematical problems based on it and many more.
Solutions from Instasolv help you to learn the elementary details of the chapter. Mean Value Theorems entail a number of theorems that are detailed geometrically and algebraically for you to learn. These solutions are provided by Instasolv’s experts in an easy language that would totally sync in with their current standard and the level of complexity. These math experts have years of experience in making the toughest of chapters in a clear and concise manner.
Topics Discussed in RD Sharma Class 12 Maths Solutions Chapter 15 Mean Value Theorems
Rolle’s Theorem –
Statement – Let a real-valued function, f has been defined on [a,b], the closed interval such that the following hold true
- It is found to be continuous on [a,b], the closed interval
- It is found to be differentiable on (a,b), the open interval
- F(a) = f(b)
If the above three hold true, then there is a real number, say s∈(a,b) such that f’(s) = 0.
Algebraic interpretation of Rolle’s Theorem
Consider c and d the roots of f(x), the polynomial. Assuming the continuity and differentiability of polynomial function with c and d being the roots of f(x), then f(c) = f(d) = 0. So f(x) will satisfy the applicability of Rolle’s Theorem.
Consequently, there would be a value e∈(c,d) resulting in f’(e) = 0 at x = e.
You can also say x = e can be considered a root of x = e.
Note – Rolle’s Theorem usually helps in the formulation of two categories of problems.
- Applicability of Rolle’s theorem can be checked on a given function in a given interval
- Rolle’s theorem can be verified on a given interval for a specific given function.
In order to solve both the problems, we will need to check if f(x) satisfies Rolle’s Theorem or not. Please consider the following results –
- Functions that are differentiable and continuous everywhere are the sine function, cosine function and the exponential function.
- Polynomial function is found differentiable and continuous everywhere
- Logarithmic Functions in their domain are differentiable and continuous
- If x=±π/2, ±3π/2, ±5π/2,…. then tan x will not be continuous
- At x=0, |x| is not differentiable
- X->k, If f’(x) tends to ±∞ then f(x) will not be differentiable at x = k.
- Operations performed on continuous (differentiable) functions such as difference, sum, quotient, and product will always produce continuous (differentiable) functions.
Lagrange’s Mean Value Theorem
Let the function f(x) defined on closed interval [a,b] so that
- F(x) remains continuous on closed interval [a,b]
- F(x) remains differentiable on the open interval (a,b)
Then, a real number c∈(a,b) will exist that would give f’(c) = (f(b)-f(a))/(b-a)
Discussions of Exercises of RD Sharma Class 12 Maths Chapter 15 Mean Value Theorems
- The first 2 exercises test your knowledge on theorems wherein you will be asked to prove a theorem’s validity. There are questions like finding the points on a given curve where the tangent will be parallel to the x-axis or checking the validity of Rolle’s Theorem in a given set of equations and many more.
- The second 2 exercises will ask you the methodology to prove the validity of Lagrange’s theorem in given conditions and inputs such as a function. You will also answer the applicability of Lagrange’s mean value theorem for function f(x) where f(x)’s value may vary from question to question.
Benefits of RD Sharma Class 12 Maths Solutions for Chapter 15 Mean Value Theorems
- In RD Sharma Class 12 Solutions by Instasolv for Chapter 15, Mean Value Theorems are designed by the best experts having extensive experience in the field.
- These solutions are curated keeping the importance of understanding the purpose of the chapter, readability, understandability, and the lack of time students have in this crucial phase of their career.
- These solutions will only add to your current understanding of the chapter, readying you for more academic challenges you’ll face in entrance examinations.