Rolle’s Theorem
Rolle’s Theorem is a special case of Lagrange’s Mean Value Theorem. It is also used to find the mean value of any function in a defined interval. It is applied when the initial value of the function f(a) equals the final value of the function f(b). For any function f(x) that is defined on the closed interval [a, b] Rolle’s Theorem is applied if,
- f(x) is continuous in the closed interval [a, b]
- f(x) is differentiable in the open interval (a, b)
- f(b) = f(a)
Then, according to Rolle’s Theorem, there exists at least one number c ∈ (a, b) such that
f'(c) = 0
where, f'(c) is the differentiation of f(x) at point c.
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