Let $f$ be a real valued function continuous on the closed interval $[0,1]$ and differentiable on the open interval $(0,1)$, with $f(0)=1$ and $f(1)=0$. Which of the following must be true?
- There exists $x \in (0,1)$ such that $f(x)=x$.
- There exists $x \in (0,1)$ such that $f'(x)=-1$.
- $f(x)>0 $ for all $x \in [0,1)$
I think the solution is
1 only, but apparently is
1 and 2 only.
I don't see why 2 is true, if the derivative of $f$ is not necessarily continuous.