let $\displaystyle f: [0,2] \to \mathbb{R} $ continuous on [0,2] and differentiable on (0,2) with $\displaystyle f(0)=f(1)= 1 \ and , f(2)= 3 $

use the mean value theorem and Darboux's theorem to prove that there is $\displaystyle c \in (0,2 ) \ with f'(c)= \frac{1}{5} $