TITLE: Prove that $c^3f(c)+c f(c)\ge1$
QUESTION [5 upvotes]: Let $f:[0,1] \rightarrow \mathbb{R}$ a continuous function that satisfies $\int_0^1 2 x f(x) dx >\pi/2$. I need to prove there is
$c\in(0,1)$ such that $c^3f(c)+c f(c)\ge1$.
My guess is the mean value theorem. Am I on the right track?

REPLY [5 votes]: Since
$$\int_0^1\big(xf(x)-\frac{1}{1+x^2}\big)dx=\int_0^1xf(x)dx-\frac{\pi}{4}>0,$$
there exists $c\in(0,1)$, such that $cf(c)-\frac{1}{1+c^2}>0$. The conclusion follows.
Remark: It suffices to assume that $f$ is (Riemann or Lebesgue) integrable on $[0,1]$. Continuity is not necessary.