Let f(x)=Ax^3+Bx^2+Cx+D and f(x) \ge 0\; ,\; \forall x\in [0,1], show that
\int_{0}^{1} xf(x)\; dx \le K\int\limits_{0}^{1}f(x)\; dx \;\;\; \; where K=\frac{6+\sqrt{6}}{10}

Any ideas? I've run out.