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Rolle's theorem
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Rolle's theorem
In real analysis, a branch of mathematics, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle.
If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.
For a radius r > 0, consider the function
![{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}},\quad x\in [-r,r].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ef1f348169f09e9db96b68d4b2e73d5f43c156)
Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function
![{\displaystyle f(x)=|x|,\quad x\in [-1,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06020ba2e58842ee0cea4e5c27566b16a4ecaff6)
Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero. This is because that function, although continuous, is not differentiable at x = 0. The derivative of f changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the open interval (a, b), but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
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Rolle's theorem
In real analysis, a branch of mathematics, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle.
If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.
For a radius r > 0, consider the function
Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function
Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero. This is because that function, although continuous, is not differentiable at x = 0. The derivative of f changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the open interval (a, b), but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
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