Putnam 1957 B3

For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that $$\int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$