Darboux Theorem

[Moubinool]

Darboux Theorem states that the derivative of a function has the Intermediate Value Property
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Darboux property.

Let E be an interval. It is said that the function f:E-->R has the property of Daboux on E, if for any a<b in E and any real number alpha situated between f(a) and f(b), there exists a least one \alpha from ]a,b[ so that f(x_\alpha)= \alpha.

thank you Lagrangia

[mr_flirt]

thanks a lot, moubi

[liyi]

[liyi]

Let's first proof this.

LEMMA f(x) is differential on [a,b]. f'(a)f'(b) < 0. there exists \xi \in (a,b) such that f'(\xi) = 0.

suppose f'(a)>0 and f'(b)<0. (the similar argument works for f'(a)<0 and f'(b)>0).

f(x) is differential on [a,b] implies that it is continuous. Thus it has a maximum value. f(a) is not the maximum value since f'(a)>0, f(x) is increasing at a. Simiarly, f(b) is not either since f(x) decreases at b.
Hence there is a \xi \in (a,b) s.t. f(\xi) is the maximum value. By Fermat's theorem, f(\xi)=0.

Now back to the general cases.

suppose c is between f'(a) and f'(b).
Let g(x) = f(x) - cx, g'(x) = f'(x) - c. g'(a)g'(b) < 0. we know from the lemma that there is a \xi such that g'(\xi) = 0, which is f'(\xi) = c.

[Moubinool]

Can someone tell me what Fermat's theorem say (maybe we have the same in french in a different name)

[liyi]

FERMAT THEOREM: suppose f(x) reaches its maximum at x_0 (x_0 is an interior point of a interval) and f'(x_0) exists, then f'(x_0) = 0.