Show that there exists no non-negative continuous function $f$ defined on the interval $[a,b]$ such that it satisfies the following conditions: $$\int_a^b f(t)dt=1 \quad \int_a^b tf(t)dt = c \quad \int_a^b t^2f(t)dt = c^2,$$ for some $c\in\mathbb{R}$.
I was given a hint that I need to use Cauchy-Schwarz inequality, and I am familiar with Cauchy-Schwarz, I just do not see how to apply it to this problem. Cauchy-Schwarz inequality states that if $u$ and $v$ are elements of an inner product space then, $\| \langle u,v \rangle \| \leq \| u \| \| v \|$.
So I guess here I have the inner product space of $C([a,b])$, continuous functions on $[a,b]$, and I have that $$\langle f, 1 \rangle = 1 \quad \langle f,t\rangle = c \quad \langle f,t^2 \rangle = c^2$$ I don't know how to piece it together though. Any help would be appreciated!