Then there is at least one point ξ (a
Rolle theorem is named after French mathematician Rolle.
Geometric meaning
The geometric meanings of the three known conditions of Rolle theorem are as follows: f(x) continuity on [a, b] indicates that the curve is seamless, including the endpoints; The derivation that (a, b) contains f(x) shows that the curve y=f(x) has a tangent at every point; F(a)=f(b) indicates that the secant (straight line AB) of the curve is parallel to the X axis. The intuitive significance of the conclusion of Rolle's theorem is that at least one point ξ can be found in (a, b), so that f'(ξ)=0, which means that the tangent slope of at least one point on the curve is 0, so that the tangent is parallel to the secant AB and also parallel to X.
Two: Rolle's theorem can be intuitively understood as: if the two endpoints of a differentiable function are the same, there must be an intermediate value with a derivative of 0. Intuitive understanding means that the function image should rise (fall) first and then fall (rise) back to the original value, and there should be a relatively flat place (not very strict, intuitive imagination). Lagrange means that the two endpoint values are different, and there is a value in the middle that can be achieved. The idea of proof is a constructor, which transforms oblique into flat (intuitive imagination).
Three: Rolle's mean value theorem:?
Set the function? F(x) is defined in the interval [a, b], if?
(1) function? F(x) is continuous on the closed interval [a, b]; ?
(2) Function? F(x) is differentiable in the open interval (a, b); ?
(3) Function? The function values of f(x) at both ends of the interval are equal, that is? f(a)=？ f(b)？
Is there at least one point in (a, b)? a & lt？ ξ? & ltb，make？ f？ ? (ξ? )=0? . ?
Geometric explanation of Rolle theorem:?
When the curve equation meets the requirements of Rolle's theorem, at least one point in the interval makes the slope of the tangent of this point zero, in other words, the tangent of this point is parallel to? x？ Axis.
[example]? Don't look for a function? The derivative of f (x) = (x-1) (x-2) (x-3) (x-4) (x-5), explain the equation? f(x)=0？ There are several real roots, and point out their intervals. ?
Solution: Because of the function? f(x)=(x- 1)(x-2)(x-3)(x-4)(x-5)？ Continuous and differentiable on the whole real axis, and? f( 1)=？ f(2)=f(3)=f(4)=？ F(5)=0, respectively in the interval? ( 1,2),? (2,3),? (3,4),? (4,5)? Using Rolle theorem, can we get the equation? f(x)=0？ There are at least four real roots, but since f(x) is a quartic polynomial with at most four real roots, the equation? f(x)=0？ There are only four real roots and they are located in the interval? ( 1,2),? (2,3),? (3,4),? (4,5)? Inside.