# Mathematics | Rolle’s Mean Value Theorem

Suppose f(x) be a function satisfying three conditions:

1) f(x) is continuous in the closed interval a ≤ x ≤ b

2) f(x) is differentiable in the open interval a < x < b

3) f(a) = f(b)

Then according to Rolle’s Theorem, there exists **at least one** point ‘c’ in the open interval (a, b) such that:

f ‘ (c) = 0

We can visualize Rolle’s theorem from the figure(1)

Figure(1)

In the above figure the function satisfies all three conditions given above. So, we can apply Rolle’s theorem, according to which there exists at least one point ‘c’ such that:

f ‘ (c) = 0

which means that there exists a point at which the slope of the tangent at that is equal to 0. We can easily see that at point ‘c’ slope is 0.

Similarly, there could be more than one points at which slope of tangent at those points will be 0. Figure(2) is one of the example where exists more than one point satisfying Rolle’s theorem.

Figure(2)

This article has been contributed by Saurabh Sharma.

If you would like to contribute, please email us your interest at contribute@geeksforgeeks.org

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: **DSA Self Paced**. Become industry ready at a student-friendly price.

## Recommended Posts:

- Mathematics | Lagrange's Mean Value Theorem
- Cauchy's Mean Value Theorem
- Quotient Remainder Theorem
- Kleene's Theorem in TOC | Part-1
- Corollaries of Binomial Theorem
- Arden's Theorem and Challenging Applications | Set 2
- Consensus Theorem in Digital Logic
- Bayes's Theorem for Conditional Probability
- Arden's Theorem in Theory of Computation
- Advanced master theorem for divide and conquer recurrences
- Mathematics | Generalized PnC Set 2
- Mathematics | Probability
- Mathematics | Generalized PnC Set 1
- Mathematics | Rules of Inference
- Mathematics | Predicates and Quantifiers | Set 2
- Mathematics | Indefinite Integrals
- Mathematics | Introduction to Proofs
- Mathematics | Law of total probability
- Mathematics | Generating Functions - Set 2
- Mathematics | Predicates and Quantifiers | Set 1