For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$ Let $f: [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1 f (t) dt =1 .$$ Show that there exists a point $c \in (0,1)$ such that $ f(c)=3c^2$.
I tried using mean value theorem of integrals on $f (x)$, so $$\int_0^1 f (t) dt = f (c) =1,$$ but it should be $3 c^2$. What is going wrong?

HINT: Let $g(x) = \int_{0}^{x}f(t)\,dt - x^{3}$. Then $g(0) = g(1) = 0$. Now apply Rolle's Theorem on $g(x)$.
