I have a Homework question which is:
Let $f(x)$ be an integrable function in $[0,1]$ and there exists a value $M>0$ such that for $x\in[0,1]$ then $f(x)>M$. Let $a=\frac{1}{2}\int_{0}^{1}{f(x)dx}.$
Prove that there is a value $c\in[0,1]$ such that $a=\int_{0}^{c}{f(x)dx}$, and that there is only one number $c$ which satisfies that condition.
I am not really sure how to solve this, I can prove that $\frac{1}{2}\int_{0}^{1}{f(x)dx}\ge M/2$. But I don't think that is the right direction.
Can someone please help me out?
Thanks :)