Proof of differentiability
Quote:
Let $\displaystyle f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.
If also f is differentiable on (0,1) prove that $\displaystyle \exists \ \theta \in (0,1)$ such that $\displaystyle \frac{f'(\theta)}{\cos \left( \frac{\pi}{2} \theta \right)}=\frac{2}{\pi}$.
I wasn't sure how to start this but I thought it had something to do with a Taylor series.
So I did this:
$\displaystyle \frac{f'(\theta)}{\cos \left( \frac{\pi}{2} \theta \right)}=\frac{2}{\pi} \Rightarrow f'(\theta)=\frac{2}{\pi} \cos \frac{\pi}{2} \theta$
$\displaystyle f'(\theta)=\frac{2}{\pi} \left( 1- \frac{ \left( \frac{\pi}{2} \theta \right)^2}{2!} + \frac{\left( \frac{\pi}{2} \theta \right)^4}{4!}-....\right)$
and I know that this is only valid for $\displaystyle \theta \in (0,1)$.
However, this isn't a proof!
Can someone help me finish this?