# Rolle’s theorem

The theorem Rolle is a proposition of the differential calculus which states that if a function of a real variable is derivable in the open interval I and continuous in the closure of I , then there is at the least one point of the range I in which the derivative is canceled. The theorem was presented by the French mathematician Michel Rolle in his Traité d’algèbre in 1690 . This property is known to have been known to the Indian mathematician Bhaskara Acharia ( 1114 – 1185 ). As known today, the theorem was proved byLouis Cauchy ( 1789 – 1857 ) as a corollary of the Mean Value Theorem (de Lagrange) of 1823.

## Summary

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• 1 Statement
• 1 Geometric interpretation
• 2 Test
• 3 References

## Statement

Let f be a function of a real variable, continuous on the closed interval [a, b] and differentiable at the interior points of said interval and f (a) = f (b). Then, there exists at least one interior point c of the closed interval [a, b] such that the derivative of f cancels out: f ‘(c) = 0 . [one]

### Geometric interpretation

When a curve y = f (x) has tangent at all its points and there are also two points with the same ordinate, then at least at one point of said curve the tangent line is parallel to the OX axis. [2]

## Proof

Since f is continuous in the compact interval [a; b], by the Weirstrass theorem there exist c and d , so that for any element x of [a; b], it is verified that f (c) <= x <= f (d) .

If f (c) = f (d) would imply that f is constant on the interval [a; b], and, thus, f ‘(e) = 0 for every element of (a; b).

Let us assume that f (c) <f (d). Since f (a) = f (b) it must happen that c or d are inside [a; b]. Let it be the case that c is in (a; b), since f has a local minimum in c and since f is differentiable, it follows that f ‘(c) = 0. [3]

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