Prove $c$ satisfies the integral If $f:[0,1] \to \mathbb{R}$ is continuous, show that there exists $c \in [0,1]$ such that
$$f(c)=\int_0^1 2t  f(t) \text{d}t.$$
So it's pretty clear to me that I have to use Intermediate Value Theorem and Cauchy-Schwarz inequality but I can't quite get the trick done.
Any help appreciated. 

Using first mean value theorem for integration (you can find here) we have that exists a $c\in\left[0,1\right]$ such that $$\int_{0}^{1}2tf\left(t\right)dt=f\left(c\right)\int_{0}^{1}2tdt=f\left(c\right).$$
