Rolle’s Theorem and Lagrange’s Mean Value Theorem
What is the mean value theorem?
Mean value theorem is a theorem which is used to find the mean of the given function. The mean value theorem formula is,
f'(c) = [f(b) – f(a)] / (b-a)
How to find c in the mean value theorem?
For any function f(x) defined on the interval [a, b] the value of c in the mean value theorem is calculated by using the formula,
f'(c) = [f(b) – f(a)] / (b-a)
What are the conditions for Lagrange Mean Value Theorem?
Condition For Lagrange Mean Value Theorem are, for any function f(x)
- f(x) should be continuous over the closed interval [a,b]
- f(x) should be differentiable over the open interval (a,b)
What are Conditions for Rolle’s Theorem?
Conditions for Rolle’s Theorem are, for any function f(x)
- f(x) should be continuous over the closed interval [a,b]
- f(x) should be differentiable over the open interval (a,b)
- f(b) = f(a)
Is Rolle’s Theorem the same as Mean Value Theorem(MVT)?
No, Rolle’s Theorem is not same as Mean Value Theorem. It is a special case of Mean Value Theorem which occurs when the value of function at initial point and the value of function at final point is same.
Rolle’s Theorem and Lagrange’s Mean Value Theorem
Rolle’s Theorem and Lagrange’s Mean Value Theorem: Mean Value Theorems (MVT) are the basic theorems used in mathematics. They are used to solve various types of problems in Mathematics. Mean Value Theorem is also called Lagrenges’s Mean Value Theorem. Rolle’s Theorem is a subcase of the mean value theorem and they are both widely used. These theorems are used to find the mean values of different functions.
Rolle’s theorem, a special case of the mean-value theorem in differential calculus, asserts that under certain conditions, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) equals f(b), then there exists at least one point x in the interval (a, b) where the derivative of f, denoted as f‘(x), equals zero.
Table of Content
- Rolle’s Theorem
- Geometric Interpretation of Rolle’s Theorem
- Proof of Rolle’s Theorem
- Example of Rolle’s Theorem
- Lagrange’s Mean Value Theorem
- Geometrical Interpretation of Lagrange’s Mean Value Theorem
- Proof of Lagrange’s Mean Value Theorem
- Application of Lagrange’s Mean Value Theorem
- Example of Lagrange’s Mean Value Theorem