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. 

Since $f$ is continuous there exist $a,b\in [0,1]$ such that $f(a)\le f(x)\le f(b), \:\forall x\in [0,1].$ Now,
$$t\in[0,1]\implies tf(a)\le t f(x)\le t f(b).$$ So
$$2f(a)\int_0^1 tdt \le 2\int_0^1 tf(t)dt\le 2f(b)\int_0^1 tdt.$$
Can you finish now?
