1. ## mean value theorem

let $f: [0,2] \to \mathbb{R}$ continuous on [0,2] and differentiable on (0,2) with $f(0)=f(1)= 1 \ and , f(2)= 3$
use the mean value theorem and Darboux's theorem to prove that there is $c \in (0,2 ) \ with f'(c)= \frac{1}{5}$

2. Originally Posted by flower3
let $f: [0,2] \to \mathbb{R}$ continuous on [0,2] and differentiable on (0,2) with $f(0)=f(1)= 1 \ and , f(2)= 3$
use the mean value theorem and Darboux's theorem to prove that there is $c \in (0,2 ) \ with f'(c)= \frac{1}{5}$
What have you done?!?! Do you even check your answers or are you some kind of hellish poltergeist who just posts random questions?

3. since f(0)=f(1), by the mean vaule theorem there exist a point s in (0,1) such that f'(s)=0,
similarly there exist a point t in (1,2) such that f'(t)=2.
thus by the Darboux's theorem, there exist a point c in (s,t) such that f'(c)=0.2 QED.

4. Originally Posted by Shanks
since f(0)=f(1), by the mean vaule theorem there exist a point s in (0,1) such that f'(s)=0,
similarly there exist a point t in (1,2) such that f'(t)=2.
thus by the Darboux's theorem, there exist a point c in (s,t) such that f'(c)=0.2 QED.
It is good practice to not answer the question after someone else asks what they have done.

5. Originally Posted by Drexel28
It is good practice to not answer the question after someone else asks what they have done.