Existence of $c$
Posted: Sat Oct 29, 2016 10:52 am
Let $f:[0, 1] \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=0$ and
\begin{equation} \int_0^1 x f(x) \, {\rm d}x = \int_0^1 f(x) \, {\rm d}x \end{equation}
Prove that there exists a $c \in (0, 1)$ such that $\displaystyle \int_0^c x f(x) \, {\rm d}x = \frac{c}{2} \int_0^c f(x) \, {\rm d}x$.
\begin{equation} \int_0^1 x f(x) \, {\rm d}x = \int_0^1 f(x) \, {\rm d}x \end{equation}
Prove that there exists a $c \in (0, 1)$ such that $\displaystyle \int_0^c x f(x) \, {\rm d}x = \frac{c}{2} \int_0^c f(x) \, {\rm d}x$.