I got the following question by mail from someone I don't know from Adam. (Quoted in part.)
if $f(t)$ continuously diff. on $[0,1]$ and
a) $\int_0^1f(t)\ dt=0$
b) m\le f\,'\le M on $[0,1]$
Prove
$\frac m{12}\le\int_0^1t\cdot f(t)\ dt\le\frac M{12}$
I suspect it might be an error
I assumed immediately that it's an error, but my first two thoughts as counterexamples were $f(t)=\frac12-t$ and $f(t)=\sin(2\pi t)$, both of which satisfy the result. Anyone with a proof or counterexample?