Mathematics : Differential calculus

Rolle's theorem

Let f(x) be a continuous function on the closed interval [a,b].

IF

- it's differentiable at every point in the interval;

- and ( f(a) = f(b) )

THEN

There is SOME point c (below: c₀, c₁, c₂, c₃) in the interval [a,b], such that the derivative of f(x) is 0 (f'(x) = 0).


Geometric meaning
If a curve is continuous, has two x-intercepts, and has a tangent at every point between the intercepts, at least one of these tangents is parallel to the x-axis (below: t₀, t₁, t₂, t₃).

: IMAGE/APPLET : Rolle's theorem, Geometric meaning
  (if applet running) Points that you can move → B, Visualizza DATI