On the integral mean value theorem

For an arbitrary function $f$ continuous on the interval $[0,1]$, the author considers a function $\xi\colon[0,1]\to R$ defined as follows. For every $x\in[0,1]$ $\xi(x)$ is the maximum of the numbers $t\in[0,x]$ satisfying the equation $f(t)x=\int_0^x f(\tau)\,d\tau$. The main result of the article consists in proving the inequality $\varlimsup\limits_{x\to0}\xi(x)/x\ge1/e$.