Rolle's theorem

Roll's theorem ( derivative zero theorem ) states that

If a real function continuous on a segment $[a,b]$ ${\ displaystyle [a, b]}$ $[a, b]$ and differentiable on an interval $(a,b)$ ${\ displaystyle (a, b)}$ $(a, b)$ , takes on the ends of the segment $[a,b]$ ${\ displaystyle [a, b]}$ $[a, b]$ the same values then on the interval $(a,b)$ ${\ displaystyle (a, b)}$ $(a, b)$ there is at least one point at which the derivative of the function is zero.

Content

Proof

If the function on the segment is constant, then the assertion is obvious, since the derivative of the function is zero at any point of the interval.

If not, since the function values at the boundary points of the segment are equal, then, according to Weierstrass , it takes its highest or lowest value at some point of the interval, that is, has a local extremum at this point, and by the Ferm lemma, the derivative at this point is 0.

Geometrical meaning

The theorem states that if the coordinates of both ends of a smooth curve are equal, then there is a point on the curve where the tangent to the curve is parallel to the x-axis.

The geometric meaning of the Roll theorem

Consequences

If a differentiable function goes to zero in $n$ ${\ displaystyle n}$ different points, then its derivative vanishes at least $n-1$ ${\ displaystyle n-1}$ different points ^[1] , and these zeros of the derivative lie in the convex hull of the zeros of the original function. This corollary is easily verified for the case of real roots, but it also takes place in the complex case.

If all the roots of a polynomial of degree n are real, then the roots of all its derivatives to $n-1$ ${\ displaystyle n-1}$ inclusive - also exclusively valid.

The differentiable function on the interval between its two points has a tangent parallel to the secant / chord drawn through these two points.

Corollary of Rolle's theorem: between every two consecutive roots of a polynomial lies the root of its derivative

Notes

↑ Bakhvalov N. S., Zhidkov N. P. , Kobelkov G. M. - Numerical Methods, p. 43

Literature

Fichtengolts GM Basics of mathematical analysis. - M .: " Science ", 1962. - T. 1. - p. 225. - 607 p.

[1] Bakhvalov N. S., Zhidkov N. P. , Kobelkov G. M. - Numerical Methods, p. 43