Someone can show me the dimonstration of the Rolle's theorem about the existence of a tangent, with derivative 0, to the f(x) function in a closed interval [a, b] with f(a)=f(b) ?

Question

alfie · Answer

Let x1 be the minimum and x2 be the maximum.

f(x1) <= f(x) <= f(x2)

Since we have a closed interval and Weierstrass theorem tells us these points do exist.

Now, if one of these two point is INSIDE of the inteval [a,b] then we have that f'(x0) = 0 (Fermat's theorem).

And the theorem is proven.

if neither of the points are in the interval, then we have that they must be a or b themselves... like..

f(a) <= f(x) <= f(b)
But f(a) = f(b) (hypotesis).
So it's a line parallel to the x-axis., where f'(x) = 0 all the time.

jamesj · Answer

@decripter, are you asking for a proof of Rolle's theorem; or are you asking for an example of it?

anonymous · Answer

@alfie, remind me Weierstrass theorem and Fermat's one please... :3
@james, I'm asking just for the dimonstration ;)