Rolle's mean value theorem is an important theorem in differential calculus and one of the three theorems of differential mean value. The other two are Lagrange mean value theorem and Cauchy mean value theorem.

Rolle's theorem is described as follows: If? r？ The function f(x) on satisfies the following conditions: In a closed interval? Continuous on [a, b], in the open interval? (a, b) is internally derivable, and f(a)=f(b), then at least one ξ∈(a, b) exists, so f'(ξ)=0.

If the arc segment AB corresponding to the continuous curve y=f(x) in the interval [a, b] has tangents that are not perpendicular to the X axis everywhere except the end points, and the vertical coordinates of the two end points A and B of the arc are equal, there is at least one point C on the arc AB, so that the tangent of the curve at the point C is parallel to the X axis.

Extended data

Practical application:

Calculus is developed with practical application, and is more and more widely used in various branches of natural science, social science and applied science such as astronomy, mechanics, chemistry, biology, engineering and economics. In particular, the invention of computers is more conducive to the continuous development of these applications.

Everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introducing the concept of variables into mathematics, it is possible to describe the movement phenomenon in mathematics.

Due to the emergence and application of the concept of function and the needs of the development of science and technology, a new branch of mathematics has emerged after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the greatest creation in all mathematics after Euclidean geometry.

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