Difference between revisions of "Rolle theorem"
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| − | If a real-valued function | + | {{TEX|done}} |
| + | If a real-valued function $f$ is continuous on some closed interval $[a,b]$, has at each interior point of this interval a finite derivative or an infinite derivative of definite sign and at the end points of this interval takes equal values, then in the open interval $(a,b)$ there exists at least one point at which the [[Derivative|derivative]] of $f$ vanishes. | ||
| − | The geometric sense of Rolle's theorem is that on the graph of a function | + | The geometric sense of Rolle's theorem is that on the graph of a function $f$ satisfying the requirements of the theorem there exists a point $(\xi,f(\xi))$, $a<\xi<b$, such that the tangent to the graph at this point is parallel to the $x$-axis. |
The mechanical interpretation of Rolle's theorem is that for any material point moving continuously along a straight line and which has returned after a certain period of time to the initial point there exists an instant at which the instantaneous velocity has been zero. | The mechanical interpretation of Rolle's theorem is that for any material point moving continuously along a straight line and which has returned after a certain period of time to the initial point there exists an instant at which the instantaneous velocity has been zero. | ||
Latest revision as of 14:12, 1 May 2014
If a real-valued function $f$ is continuous on some closed interval $[a,b]$, has at each interior point of this interval a finite derivative or an infinite derivative of definite sign and at the end points of this interval takes equal values, then in the open interval $(a,b)$ there exists at least one point at which the derivative of $f$ vanishes.
The geometric sense of Rolle's theorem is that on the graph of a function $f$ satisfying the requirements of the theorem there exists a point $(\xi,f(\xi))$, $a<\xi<b$, such that the tangent to the graph at this point is parallel to the $x$-axis.
The mechanical interpretation of Rolle's theorem is that for any material point moving continuously along a straight line and which has returned after a certain period of time to the initial point there exists an instant at which the instantaneous velocity has been zero.
This theorem was first obtained by M. Rolle [1] for algebraic polynomials.
References
| [1] | M. Rolle, "Traité d'algèbre" , Paris (1690) |
| [2] | S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) |
Comments
See also Finite-increments formula.
References
| [a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff |
| [a2] | T.M. Apostol, "Calculus" , 1 , Blaisdell (1967) |
Rolle theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rolle_theorem&oldid=13403