Регистрация Вход
Математика 10-11 класс Syed Waqas Ali

Q: State And Prove Rolles Theorem With Diagram?

0
Ответы:

Rolle's theorem

Theorem. Let the function y=f(x) be continuous on the interval [a,b] , have a derivative f’(x) on the interval (a,b) and, at the same time, f(a)=f(b). Then there is a point c∈ (a,b) at which the condition f'(c)=0 is satisfied.

Proof. The function y=f(x) is continuous on the segment [a,b] and, therefore, reaches its maximum and minimum values ​​on this segment. If these values ​​are the same, then the function is equal to a constant, and its derivative is equal to 0 at each point of the interval (a,b). If the largest and smallest values ​​of the function do not coincide, then at least one of them does not coincide with the value of the function on the boundaries of the segment. Let at a point c∈ (a,b) the maximum or minimum value of the function on the interval be reached. Then this point is an extremum point and at this point, according to Fermat's theorem, the derivative is equal to 0.

Geometric interpretation. The theorem means that if the function y=f(x) satisfies Rolle's theorem, then there is at least one point C such that the tangent to the graph of the function drawn at this point is parallel to the axis Ox.

Consequence. If f(a)=f(b)=0, then Rolle's theorem can be formulated as follows: between two successive zeros of a continuous differentiable function there is at least one zero of the derivative.

1
Отв. дан Truess
Для написания вопросов и ответов необходимо зарегистрироваться на сайте

Другие вопросы в разделе - Математика

Syed Waqas Ali
Syed Waqas Ali
Yxanovmisha Com1
Yxanovmisha Com1
Yxanovmisha Com1
Yxanovmisha Com1
Kirill
Kirill